Difference between revisions of "Zu Chongzhi" - New World Encyclopedia

Latest revision as of 06:13, 13 June 2023

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 $\textstyle {365{9589 \over 39491}}$ (≒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.

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.

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 $\textstyle {365{9589 \over 39491}}$ (≒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 ³⁵⁵⁄₁₁₃ (密率, Milu, detailed approximation) and ²²⁄₇ (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

Traditional Chinese:

指南車

Simplified Chinese:

指南车

Transliterations
Mandarin
- Hanyu Pinyin:	zhi3 nan2 che1
Yue (Cantonese)
- Jyutping:	zi2 naam4 ce1

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, ²²⁄₇ and ³⁵⁵⁄₁₁₃ 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

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

References
ISBN links support NWE through referral fees

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

External links

All links retrieved June 13, 2023.

Zu Chongzhi at Encyclopedia Britannica

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[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 ${\frac {52163}{16604}}=3.1415923874$ .

[needham_volume_4_part_2_289-3] Needham, Volume 4, Part 2, 289.

[needham_volume_4_part_2_298-4] 4.0 ^4.1 ^4.2 Needham, Volume 4, Part 2, 298.

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
+{{Images OK}}{{submitted}}{{approved}}{{Copyedited}}
 [[Image:Juchungzi.jpg|thumb|200px|Zhu Chung Zih bronze staute. Partly damaged.]]
 '''Zu Chongzhi''' ({{zh-tspw|t=祖沖之|s=祖冲之|p=Zǔ Chōngzhī|w='''Tsu Ch'ung-chih'''}}, 429–500), [[courtesy name]] '''Wenyuan''' (文遠), was a prominent [[China|Chinese]] [[List of mathematicians|mathematician]] and [[List of astronomers|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 <math>\textstyle{365{9589\over39491}}</math> (≒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]].
+{{toc}}
+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 [[gear]]s.
+{{ChineseText}}
 ==Life and works==
 Zu Chongzhi's ancestry was from modern [[Baoding|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|Zhenjiang, Jiangsu]]), he was engaged in work at the office of the local governor.
+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|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]].
+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.
+*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 <math>\textstyle{365{9589\over39491}}</math> (≒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.
+*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 nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with [[Milü|<sup>355</sup>⁄<sub>113</sub>]] (密率, Milu, detailed approximation) and [[Proof that 22 over 7 exceeds π|<sup>22</sup>⁄<sub>7</sub>]] (約率, Yuelu, rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 (= 2<sup>12</sup> &times; 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, "<math>\tfrac{22}{7}</math> was nothing more than the π value obtain several hundred years earlier by the Greek mathematician [[Archimedes]],however Milu <math>\pi=\tfrac{355}{113}</math>  could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 [[Netherlands|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 <math>\tfrac{355}{113} </math> be named after Zu Chongzhi as Zu Chongzhi fraction<ref>Yoshio Mikami: Development of Mathematics in China and Japan p50 1913</ref>. 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 16600<ref>The next "best rational approximation" to π is <math>\frac{52163}{16604}=3.1415923874 </math>.</ref>.
+*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 [[Milü|<sup>355</sup>⁄<sub>113</sub>]] (密率, Milu, detailed approximation) and [[Proof that 22 over 7 exceeds π|<sup>22</sup>⁄<sub>7</sub>]] (約率, Yuelu, rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 (= 2<sup>12</sup> &times; 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, "<math>\tfrac{22}{7}</math> was nothing more than the π value obtain several-hundred years earlier by the Greek mathematician [[Archimedes]],however Milu <math>\pi=\tfrac{355}{113}</math>  could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 [[Netherlands|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 <math>\tfrac{355}{113} </math> be named after Zu Chongzhi as Zu Chongzhi fraction.<ref>Yoshio Mikami, ''The Development of Mathematics in China and Japan.'' (New York: Chelsea Pub. Co, 1913), 50</ref> 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.<ref>The next "best rational approximation" to π is <math>\frac{52163}{16604}=3.1415923874 </math>.</ref>
 *finding the volume of a sphere as πD<sup>3</sup>/6 where D is diameter (equavilent to 4πr<sup>3</sup>/3</small>).
-*discovering the [[Cavalieri's principle]], 1000 years before [[Bonaventura Cavalieri]] in the West.
+*discovering the [[Cavalieri's principle]], 1,000 years before [[Bonaventura Cavalieri]] in the West.
-== Astronomy ==
-Zu was an accomplished astronomer who calculated the values of time to almost pinpoint precision. His methods of interpolating and the usage of integration is far ahead of his time. Even the astronomer's [[Yi Xing]] isn't comparable to his value(whom 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 was so advance, the scholars of the Sung dynasty and Indo influence astronomers of the Tang dynasty found it confusing.
 == 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 8 decimal places. No mathematician since his time, computed a value this precise until another 1000 years. Zu also worked on deducing the formula for the volume of the sphere.
+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==
+{{Chinese|pic=|picsize=|piccap=|t=指南車|s=指南车|p=zhi3 nan2 che1|j=zi2 naam4 ce1}}
+[[Image:The Chinese-children instruction vehicle of Expo 2005 Aichi Japan.jpg|Reconstruction of a South Pointing Chariot, 2005|thumb]]
+[[Image:South-pointing chariot (Science Museum model).jpg|thumb|right|Model in the [[Science Museum (London)|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 (mechanical device)|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]], [[Taoism|Toaist]] immortal saint.
-The [[South Pointing Chariot]] device was first invented by the Chinese mechanical engineer [[Ma Jun]] (c. 200-265 C.E.). It was a wheeled vehicle that incorporated an early use of [[differential (mechanical device)|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 [[magnetic]]s (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:
+This effect was achieved not by [[magnetic]]s (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:
 <blockquote>
 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.<ref name="needham volume 4 part 2 289">Needham, Volume 4, Part 2, 289.</ref>
 </blockquote>
+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.<ref name="needham volume 4 part 2 298">Needham, Volume 4, Part 2, 298.</ref> Joseph Williamson used a differential for correcting the [[equation of time]] for a [[clock]] that displayed both mean and [[solar]] time.<ref name="needham volume 4 part 2 298"/> 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).<ref name="needham volume 4 part 2 298"/>
 ==Named for him==
-* <math> \pi=\tfrac{355}{113} </math> as [[Milü|Zu Chongzhi <math>\pi</math> rate]]. Zu Chongzhi computed ''π'' to be between 3.1415926 and 3.1415927 and gave two approximations of ''π'', <sup>22</sup>⁄<sub>7</sub> and  <sup>355</sup>⁄<sub>113</sub> in the 5th century.<math> \pi=\tfrac{355}{113} </math> as [[Milü|Zu Chongzhi <math>\pi</math> rate]].
+* <math> \pi=\tfrac{355}{113} </math> as [[Milü|Zu Chongzhi <math>\pi</math> rate]]. Zu Chongzhi computed ''π'' to be between 3.1415926 and 3.1415927 and gave two approximations of ''π'', <sup>22</sup>⁄<sub>7</sub> and  <sup>355</sup>⁄<sub>113</sub> in the fifth century.<math> \pi=\tfrac{355}{113} </math> as Zu Chongzhi <math>\pi</math> rate.
 *The lunar crater [[Tsu Chung-Chi (crater)|Tsu Chung-Chi]]
 *[[1888 Zu Chong-Zhi]] is the name of [[asteroid]] 1964 VO1.
@@ Line 49: / Line 56: @@
 {{Reflist}}
 ==References==
+*Du, Shiran and Shaogeng He. [http://203.72.198.245/web/Content.asp?ID=43669&Query=1 "Zu Chongzhi"]. ''[[Encyclopedia of China]]'' (Mathematics Edition), 1st ed. Retrieved February 24, 2009.
-*Needham, Joseph (1986). ''Science and Civilization in China: Volume 4, Part 2''. Taipei: Caves Books, Ltd.
+*Lloyd, G. E. R. ''Principles and Practices in Ancient Greek and Chinese Science.'' Aldershot, Hampshire, Great Britain: Ashgate/Variorum, 2006. ISBN 9780860789932
-*Du, Shiran and He, Shaogeng, [http://203.72.198.245/web/Content.asp?ID=43669&Query=1 "Zu Chongzhi"]. ''[[Encyclopedia of China]]'' (Mathematics Edition), 1st ed.
+*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.
-==Further reading==
+*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 (1986). ''Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth''. Taipei: Caves Books, Ltd.
+*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
 ==External links==
-*[http://www.britannica.com/eb/article-9384156/Zu-Chongzhi Encyclopedia Britannica's description of Zu Chongzhi]
+All links retrieved June 13, 2023.
-*[http://www.chinaculture.org/gb/en_aboutchina/2003-09/24/content_26294.htm Zu Chongzhi at Chinaculture.org]
+*[http://www.britannica.com/eb/article-9384156/Zu-Chongzhi Zu Chongzhi at Encyclopedia Britannica]
-*[http://hua.umf.maine.edu/China/astronomy/tianpage/0014ZuChongzhi9296bw.html Zu Chongzhi at the University of Maine]
-*{{MacTutor Biography|id=Zu_Chongzhi}}
-{{ChineseText}}
-[[Category:Ancient mathematicians]]
+[[Category:mathematics]]
-[[Category:Chinese astronomers]]
+[[Category:Astronomy]]
-[[Category:Chinese mathematicians]]
+[[Category:engineering]]
-[[Category:5th century mathematicians]]
+[[Category:history and biography]]
-[[Category:Liu Song]]
-[[Category:Southern Qi]]
-{{credits|Zu_Chongzhi|253446281|Milü|256982947|South_Pointing_Chariot|264894864|}}
+{{credits|Zu_Chongzhi|253446281|Milü|256982947|South_Pointing_Chariot|264894864|Liang_Dynasty|238820632}}