# Zu Chongzhi

Zhu Chung Zih bronze staute. Partly damaged.

Zu Chongzhi (Traditional Chinese: 祖沖之; Simplified Chinese: 祖冲之; Hanyu Pinyin: Zǔ Chōngzhī; Wade-Giles: Tsu Ch'ung-chih, 429–500), courtesy name Wenyuan (文遠), was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.

China is one of the countries which had the most advanced mathematics before fourteenth century. Zu Chongzhi is known for his accurate approximation for π for the following 900 years. His best approximation was between 3.1415926 and 3.1415927 (355/113). Zu also calculated one year as $\textstyle{365{9589\over39491}}$ (≒365.24281481) days, which is close to today's 365.24219878 days. Zu also developed the Daming calendar (大明曆) in 465, and his son completed his work. It became the official calender of Ming Dynasty.

## Contents

Chinese mechanical engineer Ma Jun (c. 200-265 C.E.) originally invented the South Pointing Chariot, a two-wheeled vehicle which was designed to constantly point south by the use of differential gears without magnetic compass. Zu Chongzhi made a major improvement to it including the adoption of new bronze gears.

 This article contains Chinese text. Without proper rendering support, you may see question marks, boxes, or other symbols instead of Chinese characters.

## Life and works

Zu Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravages 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 "Minister of Great Works" (大匠卿) within the Liu Song and was in charge of government construction projects. Zu's father, Zu Shuo (祖朔) 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.

## Zhui Shu

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

His mathematical achievements included:

• the Daming calendar (大明曆) introduced by him in 465. His son continued his work and completed the calender. The Daming calender became official calender of Liang Dynasty (梁朝; Pinyin: Liáng cháo) (502-557).
• 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 $\textstyle{365{9589\over39491}}$ (≒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, which held as the most accurate approximation for π for over 900 years. His best approximation was between 3.1415926 and 3.1415927, with 355113 (密率, Milu, detailed approximation) and 227 (約率, Yuelu, rough approximation) 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 obtain several-hundred years earlier by the Greek mathematician Archimedes,however Milu $\pi=\tfrac{355}{113}$ 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 extraodinary 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 Chongzhi fraction.[1] In Chinese literature, this fraction is known as "Zu rate." Zu rate is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator less than 16,600.[2]
• finding the volume of a sphere as πD3/6 where D is diameter (equavilent to 4πr3/3).
• discovering the Cavalieri's principle, 1,000 years before Bonaventura Cavalieri in the West.

## Mathematics

Most of Zu's great mathematical works, are recorded in his lost text Zhui Shu. Most scholars argue about his complexity. Since traditionally, the Chinese developed mathematics as algebraic, and equational. Logically, scholars assume that his work, Zhui Shu yields methods of cubic equations. His works on the accurate value of pi describes the lengthy calculations. Zu used the method of exhaustion, inscribing a 12,288-gon. Interestingly, Zu's value of pi is precise to eight decimal places. No mathematician since his time, computed a value this precise until another 900 years. Zu also worked on deducing the formula for the volume of the sphere.

## The South Pointing Chariot

Zu Chongzhi
Simplified Chinese: 指南车
Reconstruction of a South Pointing Chariot, 2005
Model in the Science Museum in London

The South Pointing Chariot device was invented by a number of engineers since antiquity in China, including Zhang Heng (CE 78–139), and Ma Jun (c. 200-265 C.E.). It was a two-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. It is a non-magnetic compass vehicle.

Although the chariot can technologically be made to point to any direction, the south was selected based upon ancient Chinese thought that the "Son-of-heaven" (天子) faces the south. In ancient Chinese thought, geographical direction is not value neutral but highly value loaded. The idea was incorporated into Feng shui, a general geographical-astronomical theory of fortune.

The literal translation of this chariot in Chinese character, "指南車," is a combination of two characters, "vehicle" (車) and "instruction" or "teaching." The character of "teaching" (指南) consists of two Characters, "pointing" (指) and "south" (南). Hence, "teaching" is expressed as "pointing to the sought." Thus, the chariot is a vehicle for a teacher or a master or Xian, Toaist immortal saint.

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 C.E., as described in the texts of the Song Shu (c. 500 C.E.) and the Nan Chi Shu, 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.[3]

Zu Chongzhi made a new improved vehicle with bronze gears for Emperor Shun of Liu Song. The first true differential gear used in the Western world was by Joseph Williamson in 1720.[4] Joseph Williamson used a differential for correcting the equation of time for a clock that displayed both mean and solar time.[4] Even then, the differential was not fully appreciated in Europe until James White emphasized its importance and provided details for it in his Century of Inventions (1822).[4]

## Named for him

• $\pi=\tfrac{355}{113}$ as Zu Chongzhi $\pi$ rate. Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 227 and 355113 in the fifth century.$\pi=\tfrac{355}{113}$ as Zu Chongzhi $\pi$ rate.
• The lunar crater Tsu Chung-Chi
• 1888 Zu Chong-Zhi is the name of asteroid 1964 VO1.

## Notes

1. Yoshio Mikami, The Development of Mathematics in China and Japan. (New York: Chelsea Pub. Co, 1913), 50
2. The next "best rational approximation" to π is $\frac{52163}{16604}=3.1415923874$.
3. Needham, Volume 4, Part 2, 289.
4. 4.0 4.1 4.2 Needham, Volume 4, Part 2, 298.

## References

• Du, Shiran and Shaogeng He. "Zu Chongzhi". Encyclopedia of China (Mathematics Edition), 1st ed. Retrieved February 24, 2009.
• Lloyd, G. E. R. Principles and Practices in Ancient Greek and Chinese Science. Aldershot, Hampshire, Great Britain: Ashgate/Variorum, 2006. ISBN 9780860789932
• Mikami, Yoshio. The Development of Mathematics in China and Japan. New York: Chelsea Pub. Co, 1913.
• Needham, Joseph. 1986. Science and Civilization in China: Volume 4, Part 2. Taipei: Caves Books, Ltd.
• Needham, Joseph. 1986. Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
• Needham, Joseph, Shigeru Nakayama, and Nathan Sivin. Chinese Science; Explorations of an Ancient Tradition. Cambridge: MIT Press, 1973. ISBN 9780262140119
• Needham, Joseph. The Grand Titration: Science and Society in East and West. London: Allen & Unwin, 1969. ISBN 9780049310056
• Temple, Robert K. G., and Joseph Needham. The Genius of China: 3,000 Years of Science, Discovery, and Invention. New York: Simon and Schuster, 1986. ISBN 9780671620288
• Volkov, Alexei. 1997. Zhao Youqin and his calculation of (Pi). Historia Mathematica. 24(3): 301.
• Zhong, Shizu. Ancient China's Scientists. Hong Kong: Commercial Press, 1984. ISBN 9789620710537