# Zu Chongzhi

Statue of Zu Chongzi in Tinglin Park in Kunshan
Zu Chongzhi
Simplified Chinese 祖冲之
Wenyuan (courtesy name)
Simplified Chinese 文远

Zu Chongzhi (429–500), courtesy name Wenyuan, was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.

## Life and works

Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravage of war, Zu's grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang (祖昌) at one point held the position of Chief Minister for the Palace Buildings (大匠卿) within the Liu Song and was in charge of government construction projects. Zu's father, Zu Shuozhi (祖朔之) also served the court and was greatly respected for his erudition.

Zu was born in Jiankang. His family had historically been involved in astronomy research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth his talent earned him much repute. When Emperor Xiaowu of Liu Song heard of him, he was sent to an Academy, the Hualin Xuesheng (華林學省), and later at the Imperial Nanjing University (Zongmingguan) to perform research. In 461 in Nanxu (today Zhenjiang, Jiangsu), he was engaged in work at the office of the local governor.

Zu Chongzhi, along with his son Zu Gengzhi (祖𣈶之),[1] wrote a mathematical text entitled Zhui Shu (綴述; "Methods for Interpolation"). It is said that the treatise contains formulas for the volume of the sphere, cubic equations and the accurate value of pi. This book didn't survive to the present day; it has been lost since the Song Dynasty.

His mathematical achievements included:

• the Daming calendar (大明曆) introduced by him in 465.
• distinguishing the sidereal year and the tropical year, and he measured 45 years and 11 months per degree between those two, and today we know the difference is 70.7 years per degree.
• calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today.
• calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today; using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
• calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
• deriving two approximations of pi, (3.1415926538932...) which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355113 (密率, milü, close proportion) and 227 (約率, yuelü, approximate proportion) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 (= 212 × 3) sided polygon. This was an impressive feat for the time, especially considering that the device Counting rods he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. Japanese mathematician Yoshio Mikami pointed out, "$\tfrac{22}{7}$ was nothing more than the π value obtained several hundred years earlier by the Greek mathematician Archimedes, however milü $\pi=\tfrac{355}{113}$ could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe". Hence Mikami strongly urged that the fraction $\tfrac{355}{113}$ be named after Zu Chongzhi as Zu's fraction.[2] In Chinese literature, this fraction is known as "Zu's ratio". Zu's ratio is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator less than 16600.[3]
• finding the volume of a sphere as πD3/6 where D is diameter (equivalent to 4πr3/3).

## Astronomy

Zu was an accomplished astronomer who calculated the values of time with unprecedented precision. His methods of interpolating and the use of integration is far ahead of his time. Even the astronomer Yi Xing's isn't comparable to his value (who was beginning to utilize foreign knowledge). The Sung dynasty calendar was backwards to the "Northern barbarians" because they were implementing their daily lives with the Da Ming Li. It is said that his methods of calculation were so advanced, the scholars of the Sung dynasty and Indo influence astronomers of the Tang dynasty found it confusing.

## Mathematics

The majority of Zu's great mathematical works are recorded in his lost text the Zhui Shu. Most scholars argue about his complexity since traditionally the Chinese had developed mathematics as algebraic and equational. Logically, scholars assume that the Zhui Shu yields methods of cubic equations. His works on the accurate value of pi describe the lengthy calculations involved. Zu used the Liu Hui's π algorithm described earlier by Liu Hui to inscribe a 12,288-gon. Zu's value of pi is precise to six decimal places and for a thousand years thereafter no subsequent mathematician computed a value this precise. Zu also worked on deducing the formula for the volume of a sphere.

## The South Pointing Chariot

The south-pointing chariot device was first invented by the Chinese mechanical engineer Ma Jun (c. 200–265). It was a wheeled vehicle that incorporated an early use of differential gears to operate a fixed figurine that would constantly point south, hence enabling one to accurately measure their directional bearings. This effect was achieved not by magnetics (like in a compass), but through intricate mechanics, the same design that allows equal amounts of torque applied to wheels rotating at different speeds for the modern automobile. After the Three Kingdoms period, the device fell out of use temporarily. However, it was Zu Chongzhi who successfully re-invented it in 478, as described in the texts of the Book of Song and the Book of Qi, with a passage from the latter below:

When Emperor Wu of Liu Song subdued Guanzhong he obtained the south-pointing carriage of Yao Xing, but it was only the shell with no machinery inside. Whenever it moved it had to have a man inside to turn (the figure). In the Sheng-Ming reign period, Gao Di commissioned Zi Zu Chongzhi to reconstruct it according to the ancient rules. He accordingly made new machinery of bronze, which would turn round about without a hitch and indicate the direction with uniformity. Since Ma Jun's time such a thing had not been.[4][5]

## Notes

1. ^ Zu Gengzhi's name contains an uncommon and obsolete graph that many setups may not support. Its composition is [日恆], approximated by the graph 暅. See gèng (𣈶) in wiktionary or the Unihan database.
2. ^ Yoshio Mikami (1913). Development of Mathematics in China and Japan. B. G. Teubner. p. 50.
3. ^ The next "best rational approximation" to π is $\frac{52163}{16604}=3.1415923874$.
4. ^ Needham, Volume 4, Part 2, 289.
5. ^ Book of Qi, 52.905

## References

• Needham, Joseph (1986). Science and Civilization in China: Volume 4, Part 2. Cambridge University Press
• Du Shiran and He Shaogeng, "Zu Chongzhi". Encyclopedia of China (Mathematics Edition), 1st ed.