Liu Hui

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A possible likeness of Liu Hui on a Chinese postage stamp
A possible likeness of Liu Hui on a Chinese postage stamp
This is a Chinese name; the family name is 劉 (Liu).

Liu Hui (traditional Chinese: 劉徽; simplified Chinese: 刘徽; pinyin: Liú Huī, fl. 3rd century) was a Chinese mathematician who lived in the Wei Kingdom. In 263 he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art.

Contents

[edit] Mathematical work

Liu was one of the first mathematicians known to leave roots unevaluated, giving more exact results instead of approximations. Along with Zu Chongzhi, they were among the greatest mathematicians of the ancient world.[1] Liu Hui expressed all of his mathematical results in the form of decimal fractions (using metrological units), yet the later Yang Hui (c. 1238-1298 AD) expressed his mathematical results in full decimal expressions.[2][3] He also provided commentary on the mathematical proof that is identical to the Pythagorean theorem of the Greek Pythagoras (c. 580 BC-500 BC).[4] Liu Hui called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".[5] In terms of the treatment of plane areas and solid figures, Liu Hui was one of the greatest contributors to 'empirical' solid geometry. For example, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge.[6] He also figured out that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid.[6]

In his commentaries on the Jiuzhang suanshu, he presented (among other things):

  • an estimate of π in the comments to chapter 1.[7] He estimated pi to 3.141024 with a 192 (= 25 × 6) sided polygon and later calculated pi as 3.14159 by using a 3072 (= 29 × 6) sided polygon.[8] He suggested that 3.14 was a good approximation. His estimation is made with a method similar to the Greek Archimedes of 250 BC who had used a 96 (= 24 × 6) sided polygon, but with a decimal fraction at 3.142704, Liu Hiu's was a bit more accurate than Archimedes' (3.1428).[8]Incidently, Luihui also obtain another value for π as 3.142704, but he commented that this value was too large, and picked the first three digits of π=3.141024 ~3.14 and put it in to a fraction form π={157 \over 50}.Nine Chapters had used the value 3 for the π formula, but Zhang Heng (78-139 AD) had previously estimated it to the square root of 10;
  • Gaussian elimination;
  • Cavalieri's principle to find the volume of a cylinder.[9]

The commentaries often include explanations why some methods work and why others do not. Although his commentary was a great contribution, some answers had slight errors which was later corrected by the Tang mathematician and Taoist believer Li Chunfeng.

Liu Hui also presented, in a separate appendix of 263 AD called Haidao suanjing or The Sea Island Mathematical Manual, several problems related to surveying. This book contained many practical problems of geometry, including the measurement of the heights of Chinese pagoda towers.[10] This smaller work outlined instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed at right angles to them".[11] With this, the following cases are considered in his work:

  • The measurement of the height of an island opposed to its sea level and viewed from the sea[11]
  • The height of a tree on a hill [11]
  • The size of a city wall viewed at a long distance [11]
  • The depth of a ravine (using hence-forward cross-bars)[11]
  • The height of a tower on a plain seen from a hill [11]
  • The breadth of a river-mouth seen from a distance on land [11]
  • The depth of a transparent pool [11]
  • The width of a river as seen from a hill [11]
  • The size of a city seen from a mountain, [11]

Liu Hui's information about surveying was known to his contemporaries as well. The cartographer and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a rectangular grid and graduated scale for accurate measurement of distances on representative terrain maps.[12]

Liu Hui provided commentary on the Nine Chapter's problems involving the building of canal and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.[13]

Although translated into English long beforehand, Liu's work has been translated into French by Guo Shuchun, a professor from the Chinese Academy of Sciences, who began this work in 1985 and took twenty years to complete it.

[edit] See also

[edit] Notes

  1. ^ Needham, Volume 3, 85-86.
  2. ^ Needham, Volume 3, 46.
  3. ^ Needham, Volume 3, 85.
  4. ^ Needham, Volume 3, 22.
  5. ^ Needham, Volume 3, 95-96.
  6. ^ a b Needham, Volume 3, 98-99.
  7. ^ Needham, Volume 3, 66.
  8. ^ a b Needham, Volume 3, 100-101.
  9. ^ Needham, Volume 3, 143.
  10. ^ Needham, Volume 3, 30.
  11. ^ a b c d e f g h i j Needham, Volume 3, 31.
  12. ^ Hsu, 90–96.
  13. ^ Needham, Volume 4, Part 3, 331.

[edit] References

  • Chen, Stephen. "Changing Faces: Unveiling a Masterpiece of Ancient Logical Thinking." South China Morning Post, Sunday, January 28, 2007.
  • Guo, Shuchun, "Liu Hui". Encyclopedia of China (Mathematics Edition), 1st ed.
  • Hsu, Mei-ling. "The Qin Maps: A Clue to Later Chinese Cartographic Development," Imago Mundi (Volume 45, 1993): 90-100.
  • Needham, Joseph & C. Cullen (Eds.) (1959). Science and Civilisation in China: Volume III, section 19. Cambridge University Press. ISBN 0-521-05801-5.
  • 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 (1986). Science and Civilization in China: Volume 4, Physics and Physical Technology, Part 3, Civil Engineering and Nautics. Taipei: Caves Books Ltd.

[edit] External links

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