Mar 15, 2013

On the day of Pi - in honor of Zu Chong Zhi (祖沖之)

Yesterday, the 14th day of March, is known as the day of Pi because of its numerical similarity with Pi, the ratio of circle circumference and its diameter:
\[ \pi = 3.14159265758979... ... \]

The Book of Sui (history record of Sui dynasty) records Zu Chong Zhi's (祖沖之) estimate of Pi, called Yuan Zhou Lü (圆周率) in Chinese [2]:

And I translate it here:
Among the prominent ancient problems, the circumference is three if the circle diameter is one. This value is not accurate. From Liu Xin (劉歆), Zhang Heng (張衡), Liu Hui (劉徽), Wang Fan (王蕃), Pi Yan Zong (皮延宗) etc., new results have been reported. The value has been updated from time to time. Late in Song (劉宋) Dynasty, south Xu state (南徐州) [4] governor Zu Chong Zhi (祖沖之), further initiated the so-called fine approximation.

Assuming the circle diameter is one zhang(丈), its circumference is less than three zhang(丈) one chi(尺) four cun(寸) one fen(分) five li(厘) nine hao(豪) two miao(秒) seven hu(忽) and greater than three zhang(丈) one chi(尺) four cun(寸) one fen(分) five li(厘) nine hao(豪) two miao(秒) six hu(忽) [1]. The true value lies between these two numbers. As an approximation, the fine approximate (密律) is that for a circle with diameter one hundred thirteen its circumference is three hundred fifty five. The rought approximate (約律) is that for a circle with diameter seven, its circumference is twenty two. Zu also provided square root and square arithmetics, together with positive and negative numbers. His method is very precise, best in arithmetic. Zu wrote a monograph called Zhui Shu (綴術) [3], is so deep that not well understood among scholars. 
These texts clearly state Zu Chong Zhi's work on Pi, as early as the 5th century. This is a great achievement in mankind history of mathematics.

[1]: 丈尺寸分厘毫秒忽 are all decimalized length units, widely used in Greater China region (CJKV).
[2]: The reader need CJK fonts to display Chinese characters properly. Literally, 圆=circle, 周=circumference, 率=ratio. It's understood that the ratio is taken with respect to the diameter or 径.
[3]: Zhui Shu can be roughly translated as numerical methods. Zhui Shu had been chosen into a collection of mathematical textbooks since Tang Dynasty (唐). It was lost after Song Dynasty (宋) (A different Song Dynasty from Zu Chong Zhi's time).
[4]: In present day Zhejiang Province, China

No comments:

Post a Comment