数理解析研究所講究録 1392 巻 2004 年 15-26 15 Why Mathematics in Ancient China? QU Anjing 曲安京 Department of Mathematics, Northwest University, Xian, 710069, China 1. Introduction Since the beginning of the last century hundreds of scholars have devoted themselves to the discipline of the history of mathematics in China. Two approaches to the history of mathematics in China have been made, namely, discovering what mathematics was done and recovering how mathematics was done, respectively. Both approaches, however, focus on the same problem, that is mathematics in history. [1] It is often said that, compared with Greek mathematics, Chinese mathematics was characterized by apractical tradition. Many scholars hold that this tradition is the fatal weakness of Chinese mathematical science, one that prevented it from developing into modern science. Some historians of mathematics have argued that certain fundamental factors of the Greek theoretical tradition, such as proofand principle, can also be detected in the Nine Chapters $ofA$rithmetic (�九章算木�, 1 century $\mathrm{B}\mathrm{C}$ ) and Liu Hui’s Annotations (\ll 文|J 徽注》, 263 $\mathrm{A}\mathrm{D}$ ). However, it seems to us that many people are still not convinced. For abetter understanding of the value of Chinese mathematics from ahistorical perspective, we need to figure out the issue of why mathematics was done in ancient civilizations. Following the topic of why mathematics, researches would be shifted, to some extent, ffom the mathematics in history to the history of mathematics. Under these circumstances, mathematics in ancient China and other old civilizations will be placed in the whole history of mathematics. The diversity of mathematics in different civilizations would make us amore distinct picture of the history of mathematics. In this article, we will try to explore the reason why apractical tradition of mathematics was chosen in ancient China. 2. The Aim of mathematical science in ancient China 2.1 Royal science” and professional scientists Royal science is the science that was controlled and encouraged by emperors. Of the exact sciences, mathematical astronomy was the only subject that attracted agreat attention of rulers in ancient and medieval China. Mathematical astronomy was an art related to the calendar-making system in ancient periods. In every dynasty, the Royal Observatory 司天遣 was an indispensable part of ffie state. Three $\sim$ The author is grateful to Prof. Michio Yano 矢野道雄教授 for his invaluable comments on an earlier draft of this article, and to Professor Hikosaburo Komatsu $\prime \mathrm{J}\backslash$ 松彦三郎教授 for his kind invitation to present this article at RIMS. Research for it was supported by the Japan Society for the Promotion of Science (JSPS, P00019). 16 kinds of expert, mathematicians $\ovalbox{\tt\small REJECT}_{\grave{\acute{\grave{7}}}^{\mathit{1}}}'$. jE, astronomers $\star \mathrm{X}\ovalbox{\tt\small REJECT}$ and astrologers $\lambda$ $\mathrm{L}_{\mathit{4}}^{\tilde{\mathrm{r}}}$., were employed as professional scientists by the emperor. Those who were called mathematicians took charge of establishing the algorithms of the calendar-ma ing system; those who were classified as astronomers dealt with the astronomical instruments and observation. Astrologers made divination of the supposed influences of the stars and planets on human affairs and terrestrial events. None of them made researches in pure mathematics. All of these professional scientists were royal officers. Biographies of the leading scholars who worked in the Royal Observatory were recorded in the official history of each dynasty. Among $\ll \mathrm{t}$ them are Zu Chongzhi $(\grave{\ovalbox{\tt\small REJECT}}\mathrm{f}\mathrm{l}\mathrm{f}\mathrm{f}\mathrm{Z}, 429- 500)$ who compiled the Daming calendar-making system $\rangle\rangle$ $\ll\lambda$ $Rfl$ ffffi in 463 $\mathrm{A}\mathrm{D}$ , Yi-xing $(^{-}\acute{\mathrm{t}}\overline{\mathrm{T}}, 683- 727)$ who compiled the Dayan calendar-making system @Jfi $\rangle\rangle$ in 724 AD and Guo Shoujing $(\mathrm{F}\beta \mathrm{E} , 1231- 1316)$ who compiled the Shoushi calendar-making system $\ll \mathrm{F}\mathrm{x}\# 1\prime Fy$$\rangle\rangle$ in 1280 $\mathrm{A}\mathrm{D}$ . On the contrary, even the most outstanding mathematicians were ordinary people, such as Liu $3^{\mathrm{r}\mathrm{d}}$ Hui ($\lambda|\mathrm{J}ffl$ , $\mathrm{f}1$ . century, see $DSB$, $\mathrm{v}.8$ , pp.418-425), Li Ye ($Li$ Chih, $+\mathrm{m}$ , 1192-1279, see $DSB$, $\ovalbox{\tt\small REJECT} f\mathrm{L}^{1}\ovalbox{\tt\small REJECT},1202$ $\mathrm{v}.8$ , pp.313-320), Qin Jiushao ($Ch’in$ Chiu-shao, -1261, see $DSB$, $\mathrm{v}.3$ , pp.249-256,), $13^{\mathrm{t}\mathrm{h}}$ $\mathrm{Z}\mathrm{h}\mathrm{u}$ $*\mathrm{t}\mathrm{E}$ Yang Hui ( $\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}$ , $\mathrm{f}1$ . century, see $DSB$, $\mathrm{v}.14$ , pp.538-546) and Shijie ($Chu$ $Sh_{i}h$ chieh, $\mathrm{f}1.13^{\mathrm{t}\mathrm{h}}$ $*_{\backslash \backslash }$ , century, see $DSB$, $\mathrm{v}.3$ , $\mathrm{p}\mathrm{p}.265\sim 271)^{[2]}$. Few facts are known about them from the historical records except their mathematical works. [3-4] 2.2 Mathematics for mathematical astronomy It is said that there were two peaks traditional Chinese mathematics. The Nine Chapters of $\mathrm{B}\mathrm{C}$ $(263 \mathrm{A}\mathrm{D})$ Arithmetic ( $\mathrm{c}$ . 1 Centuyy ) and its Annotations by Liu Hui featured the first one. Qin J., Li $\mathrm{Y}$ , Yang H. and Zhu S. belonging to the same generation fomed another splendid peak in the 1 $3^{\mathrm{t}\mathrm{h}}$ century. Basic topics of mathematics related to civil life were main theme in time ofthe Nine Chapters and Liu Hui. But in the $12^{\mathrm{t}\mathrm{h}}$ and $13^{\mathrm{t}\mathrm{h}}$ centuries, mathematics was highly developed for the purpose of application mathematics to astronomy. For instances: Indeterminate problem. Found in Qin Jiushao’s book, the Chinese Remainder Theorem is an algorithm for dealing with aset linear congruence. It is related with the problem to seek for the superior epoch – aspecial moment when the earth, the moon and the five planets gather near the point of winter solstice, and usually with other conditions. Aset of linear congruence to find such an epoch was established. Numerical solution ofalgebraic equations, known as Ruffini-Horner method. In order to transform the length of arc on acircle to its corresponding chord, apolynomial equation of the $4^{\mathrm{t}\mathrm{h}}$ order was established by Guo Shoujing in the Shoushi calendar-making system $(1280 \mathrm{A}\mathrm{D})$ . It was the highest order algebraic equation that appeared in traditional Chinese mathematical astronomy. Interpolation series summation. The formula for higher order series summation is found in Zhu Shijie’s work. Series summation is related with finite differences. Interpolation ffinctions from $\mathrm{L}\mathrm{i}\mathrm{u}$ Zhuo’s $\lambda^{\mathrm{g}}\mathrm{E}$ quadratic $(600 \mathrm{A}\mathrm{D})$ to Guo Shoujing’s cubic $(1280 \mathrm{A}\mathrm{D})$ were constructed by finite differences. 2.3 Mathematical astronomy under imperial authorit 17 Compilation and promulgation of calendar symbolized the imperial authority in ancient China. The calendar of adynasty had to be replaced by the new dynasty. Ordinary people were strictly prohibited to construct acalendar-making system. They were not even allowed to acquire the knowledge of mathematical astronomy. The technical details of calendar-making were kept secret to public, with the result that nobody knew how atraditional Chinese calendar-making system was compiled after the tradition was discontinued by the impact of Jesuit science in the Ming and Qing dynasties (16-19 century). Calendar-makers were asked to maintain ahigh precision in prediction. It was judged by the calculated positions of the celestial bodies and the predictions of astronomical phenomena such as eclipse. It often happened that acalendar-making system was replaced by another one simply because it failed to predict asolar eclipse. In the past 2000 years, more than 100 systems appeared. Competition among calendar-makers did not concern cosmic or geometric model but numerical method. The former might tell one how acelestial body moves, the topic that people did not really care about. People were interested in the latter simply because it could tell them where a celestial body was. Thus few people paid much attention to cosmic model building except some of those who worked at the Royal Observatory. In ancient China, most mathematicians were trained as calendar-makers. Mathematics, except those which astrologers were interested in, was never apart of philosophy. It was developed for mathematical astronomy besides for ordinary application. The aim of calendar-makers and mathematical astronomy was accurate prediction but not cosmic model building. In this way, apractical tradition to mathematics was formed in ancient China. 3. Practical tradition: evolution of numerical method In his Qiarrxiang calendar-making system ( $\not\in\ovalbox{\tt\small REJECT}$ ffi, $206\mathrm{A}\mathrm{D}$), linear interpolation was employed by Liu Hong $\grave{7}$ uu to calculate the equation of center of the moon after the irregular lunar motion was discovered in the 2nd century. His method made the change of the moon’s velocity from alevel line to astep like pattern. $6^{\mathrm{t}\mathrm{h}}$ By the end of the century, after Zhang Zixin $\#\neq \mathrm{P}_{\overline{\mathrm{R}}}$ found that the apparent motion of the sun was irregular, linear interpolation was applied to the calculation of the equation of center of the sun in several calendar-making systems. The piecewise linear interpolation made the mean solar motion presented by alevel line be replaced by 24 step like lines in atropical year that was divided into 24 parts $(q\iota)$ . It was Liu Zhuo (in 600 $\mathrm{A}\mathrm{D}$ ) who changed the pattern of the sun’s apparent velocity from step like lines to aslant line.