数理解析研究所講究録
第 1648 巻 2009 年 216-220

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On Characteristic Features of Traditional
Mathematics in Ancient China
Wu Wen-tsun1

MMKL, Academy of Mathematics and System Sciences, Academia Sinica

Beijing 100080, P.R. China

Abstract
The traditional mathematics in ancient China had some characteristic featuresquite different from those represented by the euclidean system of mathematics inancient Greece which governs the development of present-day modem mathematics.
For example, for euclidean system one usually develops a logical system of reasoning based on a set of axioms admitted to be true $ad$ hoc. On the other hand
the traditional mathematics of ancient China is interested in the solving of problems
arising ffom practical needs. For this purpose, on analyzing the problems one introduces new novel concepts, arrives at methods of solving, and then formulates generalprinciple to treat further new more delicate problems. This will run recurrently andindefinitely as shown in the diagram below:
Practical Problems$\downarrow$

Concepts and Methods of Solving$\downarrow$

General Principle$\downarrow$

Further Examples$\downarrow$

As an illustration we have the diagram below:
1taken from KyotoConference in Kyoto, March 7-8,2006.

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Rectangular Area Determination$\downarrow$

Methods$\downarrow$

Out-In Complementary Principle$\downarrow$

Polygonal Area Determination$\downarrow$

The most important mathematics classics of ancient China is $\ll Nine$ Chapterscompleted in early years of Western Han Dynasty (206-8 B.C.)
of Arithmetic
by some officers in charge of economic reconstmctions. The Out-In-Complementary
Principle is explitely formulated in Chap.9 of $\ll NineChapters\gg$ in dealing withthe simple practical problem of determining the area of rectangular agricultural fields.
The Principle thus formulated permits to solve quite a lot of intricate problems including the proof of the Sun-Height Formula dated in quite remote time. Moreover,
the recurrent application of the above diagram permits the discovery of further principles: the 2:1 Principle of LIU Hui, and the LIU-ZU Principle of LIU Hui and ZU
Geng (equivalent to much later re-discovered Cavalieri Principle). These Principles
had permitted the Chinese ancient scholars to deal with great success such delicateproblems like the computation of , the determination of volumes of polyhedras andcurved solids like spheres, circular pyramids, etc.
In the 8th Chapter of $\ll Nine$ Chapters
it is shown how to solve by somedefinite elimination procedure an arbitrary system of linear equations in integer coefficients. For this sake positive and negative hactional numbers together with rulesof computations are introduced giving thus the complete system of rational numbers.
Furthermore, in order to solve such problems of determining the side of a square fieldof known integral-valued area or that of a cubic-form container with known integralvalued volume, irrational numbers of square or cubic root type are introduced witha special terminology of Mian literally meaning side. Approximate integral values ofsuch roots are shown how to compute and even some general formulae goveming such
irrationals are given in that classic.
Besides the classic $\ll Nine$ $Chapters\gg$ appeared in year 263 A.D. toward theend of the period of Three Kingdoms (220-265 A.D.) appeared the extremely important classic $\ll Annotations$ to Nine Chapters
due to scholar LIU Hui. Amongits diverse important contributions to mathematics we may cite in particular thefollowing one. By introducing the notion of decimal numbers to arbitary stage aswell as limit concepts and approximating procedure LIU Hui had actually completed
the Real Number System. This system was so introduced in a natural and easilyhandled way together with the important concept of decimal numbers. For this wemay compare the much later introduced real number systems in 19th century bywestem mathamaticians in ways quite involved and seeminly not quite natural.
$\gg$

$\pi$

$\gg$$\gg$

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A further characteristic feature of China’s ancient traditional mathematics is thefollowing one: In contrast to euclidean tradition of laying emphasis on theoremproving, the traditional mathematics of ancient China concentrated its efforts tosolving of problems arising from practical needs. As the datas given and the numericalresults to be found of the problems should be related by some kind of equations,naturally and usually in the form of polynomial ones, so polynomial equations-solving
becomes a central theme throughout the development of China’s ancient mathematics.
Thus, in the classic $\ll Nine$ Chaptersbesides the square- and cubic root
extraction corresponding to the solving of equations $x^{2}=A$ or $x^{3}=A$ in integers
fora positive integer, there appears also the solving of such quadratic equations
like $x^{2}+34*x=71000$ with solution $x=250$ , (Ch.9, Prob.20). In fact, somecontemporain scholar ZHAO Shuang of LIU Hui had written some short essays aboutmathematics. One essay concerns a detailed analysis of the Sun-Height Formula anda second one deals with the general quadratic equation. Among the results involved
we may cite that one which gives a general formula about the roots of a quadratic
equation in terms of its coefficients, which is same as the one well-known to us in
now-a-days.
The solving of polynomial equations developed incessantly up to the end of Yuan
Dynasty (1271-1386 A.D.). Thus, in the time of Southern-Northem Dynasties (420589 A.D.), there appeared the important $classic\ll ZhuiShus\gg$ (literally meaning$\ll The$ Art of Mending
due to the celebrated great mathematician and astronomer ZU Chongzhi. It is very likely a collected work of ZU (perhaps together
with his son ZU Geng $(fl. 5c A.D.)$ , also a great mathematician with invaluablecontributions). ZU Chongzhi, besides being a great scientist, is also the important
Daming-Calender maker, reknown for some revolutionary concepts and extremelyhigh accuracy in numerical datas. Moreover, ZU is also an engineer and instrumentmanufacturer. Thus, he had built some Constantly-Southem-Directed Vehicle andsome Thousand-Li-Ship. In mathematics ZU is best known for his computation ofthe value of with
$\gg$

$A$

$\gg$

$\pi$

$\frac{22}{7}<\pi<\frac{355}{113}$.

Besides, ZU had considered the problem of solving such cubic equations of the form

$x*(x+k)*(x+l)=V$, or$x^{3}+(k+l)*x^{2}+k*l*x=V$.

In the early years of Tang Dynasty (618-907 A.D.) some scholar WANG Xiaotong (fl. 626 A.D.) wrote the classic $\ll Continuation$ of Ancient Mathematics
devoting to cubic equations closely connected with complicate reconstructions. Wangseriously criticized the work of ZU on cubic equations as completely absurd. In year
656 A.D. of Tang Dynasty it was established some kind of official school for thetraining of mathematicians with ten classics as textbooks among which are the$\gg$$\ll$

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Nine $Chapters\gg$ , the $\ll Sea-$ Island $Manuel\gg$ of LIU Hui, the $\ll ZhuiShus\gg$of ZU Chongzhi, etc. However, owing to the extreme difficulties of ZU’s classic it wascompelled to be removed from the study. Henceforth this classic was lost in China andno one can tell what are its contents. In view of the well-known computational resultsof ZU about but little known about the way that ZU reached such elegant results,scholars in later years usually held the opinion that the difficulties of ZU’s classic arecaused by difficulties for the understanding of ZU’s method of such computations of. The present author cannot agree with such opinions. In fact, the principle andmethod of computing was laid down by LIU Hui in his $\ll Annotations\gg$ and LIU
himself had shown in stepwise details how to carry out the computations up to certain
degree. The computations of ZU follows closely the steps of LIU. Thus, no matter
how ingenious and how somewhat new original artifice created $ad$ hoc are introduced
by ZU for his computations, it is highly unbelievable that such artifices are so difficult
to be non-understandable by scholars of contemporain and later generations.
In the present author’s opinion, what caused ZU’s classic to be non-understandable
and to be seriously criticized by WANG Xiaotong to be absurd is this: ZU had,through his study of cubic equations, introduced some kind of new numbers equivalent
to the present-day complex numbers!!!
Remark that in Europe the complex numbers was not introduced through thestudy of such simple equation $x^{2}=-1$ , but through the trial of solving some kind ofcubic equations as in the case of G.Cardano (1501-1576 A.D.)!!!
The development of Chinese traditional mathematics reached a climax in Song
Dynasty (960-1279). In year 1247 A.D. the scholar QIN Jiushao wrote the gigantic$classic\ll Mathematical$ Treatise in Nine $Chapters\gg$ . In this classic besides various contributions including the now-a-days well-known Chinese Remainder Theorem,
we may cite in particular the following one: It concems with a general method ofsolving numerically a polynomial equation of arbitrarily high degree with arbitrary
numerical coefficients. Moreover, in the period of Southem Song (1127-1279 A.D.)
and Yuan Dynasty (1271-1368), mathematicians introduced the novel concepts ofHeaven’s Element, Earth’s Element, etc., which correspond to unknowns $x,$ , etc. inthe present-day mathematics. With the aids of such concepts the scholars in thesedays had introduced the notions of rational functions, the algebrization of geome$tw$ , and the way of establishing equations from given geometrical conditions, etc. In
particular, it leads the way to the solving of systems of polynomial equations. Suchdevelopments reached the climax in the hand of some scholar ZHU Shijie (fl. 1299A.D.). In fact, in the classic $\ll Jade$ Mirror of Four $Elements\gg$ Zhu gave a general method of solving up to 4 polynomial equations in 4 unknowns. To simplify thematter let us consider the case of 3 polynomial equations $P_{i}(x_{1}, x_{2}, x_{3})=0$ in 3 unknowns
with known numerical coefficients. Let us arrange the 3 unknowns. Let us eliminate
from pairs of equationsin an arbitrary order say$P_{i}=0$ to get a set of polynomial equations inalone: $Q_{j}(x_{1}, x_{2})=0,j=1,2,$ .
$Q_{j}=0$ to get a set of polyNext, let us eliminate
equationspairwise from these
$\pi$

$\pi$

$\pi$

$y$

$x_{1},$ $x_{2},$ $x_{3}$$x_{1}\prec x_{2}\prec x_{3}$$x_{i}$$x_{3}$$x_{1},$ $x_{2}$$x_{2}$$\cdots$

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alone: $R_{k}(x_{1})=0,$ $k=1,2,$
nomial equations ineach of the above set of equations, say:$x_{1}$$R_{1}(x_{1})=0,$$Q_{1}(x_{1}, x_{2})=0,$$\cdots$. Let us now choose one from

$P_{1}(x_{1}, x_{2}, x_{3})=0$.

Let us now solve the polynomial equation $R_{1}=0$ forwhich we know how to do byvarious known methods, e.g. that of QIN Jiushao. Substitute this value of, then substitute the values ofin $Q_{1}=0$ and solve it forto get a solution
to get solution .
thus found in $P_{1}=0$ and solve it for the sole unknownof the original given
Combining together, we get a solution
set of equations $P_{i}=0,$ $i=1,2,3$ . From the above we see that the line of thought andthe procedure is quite clear and it is available for an arbitrary number of variables
even equations. In the latter part of our article we shall exploit it in more details
together with its variety of applications.
We leave as ide other contributions of our ancient traditional mathematics, but weshould point out the fact below: In contrast to the modern mathematics inheritedfrom the ancient Euclidean tradition for which the results are expressed in the formof theorems, our traditional mathematics expresses its results usually in the form ofshui, literally meaning (stepwise) method of solving, which corresponds in essence tothe present-day algorithm.
$x_{1}$$x_{1}=x_{1}^{0}$$x_{2}$$x_{1}^{0},$$x_{2}^{0}$$x_{2}=x_{2}^{0}$$x_{3}$$(x_{1}, x_{2}, x_{3})=(x_{1}^{0}, x_{2}^{0}, x_{3}^{0})$$x_{3}^{0}$