List of Chinese discoveries

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Aside from many original inventions, the Chinese were also early original pioneers in the discovery of natural phenomena which can be found in the human body, the environment of the world, and the immediate solar system. They also discovered many concepts in mathematics. The list below contains discoveries which found their origins in China.


Imperial China[edit]

Han Dynasty (202 BC – 220 AD) paintings on tile of Chinese guardian spirits representing 11 pm to 1 am (left) and 5 am to 7 am (right); the ancient Chinese, although discussing it in supernatural terms, acknowledged circadian rhythm within the human body
The frontispiece to Hu Sihui's Principles of Correct Diet published in 1330 (Yuan Dynasty); the caption reads "Many diseases can be cured by diet alone," a belief which spanned back to at least the 3rd century AD in China.[7]
  • Deficiency diseases, correction by proper diet: As early as the 4th century BC, Warring States period (403–221 BC), records indicate that Imperial Dieticians were appointed at royal courts.[7] The first explicit description of a regulated diet used to curb certain diseases is found in the Systematic Treasury of Medicine written by Zhang Zhongjing (c. 150 – c. 219) during the late Han Dynasty.[7] Although Zhang did not understand the true nature of vitamins, he prescribed foods now known to be rich in certain vitamins, which were discovered to be useful after much trial and error.[7] The Yuan Dynasty (1271–1368) physician and Imperial Dietician Hu Sihui (fl. 1314–1330) published his book Principles of Correct Diet which compiled a large amount of previous material written on the subject.[7]
  • Diabetes, recognition and treatment of: The Huangdi Neijing compiled by the 2nd century BC during the Han Dynasty identified diabetes as a disease suffered by those who had made an excessive habit of eating sweet and fatty foods, while the Old and New Tried and Tested Perscriptions written by the Tang Dynasty physician Zhen Quan (died 643) was the first known book to mention an excess of sugar in the urine of diabetic patients.[8][9] While his book is now lost, quotations of it were preserved in the Important Medical Formulae and Prescriptions Now Revealed by the Governor of a Distant Province, written by Wang Tao in 752.[8] The Tang physician Sun Simiao (581–682) wrote in his Thousand Golden Remedies of 655 that for diabetic patients "three things must be renounced, wine, sex, and eating salted, starchy cereal products; if this regimen can be observed, cure may follow without drugs."[10] Robert Temple writes that this is similar to the modern method of avoiding alcohol and starchy foods.[10] The sweetness of urine in diabetic patients is also noted in an ancient text of India, but unlike the Chinese texts its date is ambiguous.[10]
Each bronze bell of Marquis Yi of Zeng (433 BC) bears an inscription describing the specific note it plays, its position on a 12-note scale, and how this scale differed from scales used by other Chinese states of the time; before this discovery in 1978,[14] the oldest known surviving Chinese tuning set came from a 3rd-century BC text (which alleges was written by Guan Zhong, d. 645 BC) with five tones and additions or subtractions of ⅓ of successive tone values which produce the rising fourths and falling fifths of Pythagorean tuning.[15]
  • Equal temperament: During the Han Dynasty (202 BC–220 AD), the music theorist and mathematician Jing Fang (78–37 BC) extended the 12 tones found in the 2nd century BC Huainanzi to 60.[16] While generating his 60-divisional tuning, he discovered that 53 just fifths is approximate to 31 octaves, calculating the difference at \tfrac{177147}{176776}; this was exactly the same value for 53 equal temperament calculated by the German mathematician Nicholas Mercator (c. 1620–1687) as 353/284, a value known as Mercator's Comma.[17][18] The Ming Dynasty (1368–1644) music theorist Zhu Zaiyu (1536–1611) elaborated in three separate works beginning in 1584 the tuning system of equal temperament; in an unusual event in music theory's history, the Flemish mathematician Simon Stevin (1548–1620) discovered the mathematical formula for equal temperament at roughly the same time (within 1 to 25 years of Zhu), yet he did not publish his work and it remained unknown until 1884; therefore, it is debatable who discovered equal temperament first, Zhu or Stevin.[19][20][21] In order to obtain equal intervals, Zhu divided the octave (each octave with a ratio of 1:2, which can also be expressed as 1:212/12) into twelve equal semitones while each length was divided by the 12th root of 2.[22] He did not simply divide the string into twelve equal parts (i.e. 11/12, 10/12, 9/12, etc.) since this would give unequal temperament; instead, he altered the ratio of each semitone by an equal amount (i.e. 1:2 11/12, 1:210/12, 1:29/12, etc.) and determined the exact length of the string by dividing it by 122 (same as 21/12).[22] The Harmonie Universelle (1636) written by Marin Mersenne (1588–1648) was the first publication in Europe outlining equal temperament, a new system of tuning that was passionately defended by J.S. Bach (1685–1750) in his Well-Tempered Clavier of 1722.[21]
  • First law of motion, partial description: The Mohist philosophical canon of the Mojing, compiled by the followers of Mozi (c. 470 – c. 390 BC), provides the earliest known attempt to describe inertia: "The cessation of motion is due to the opposing force...If there is no opposing force...the motion will never stop. This is as true as that an ox is not a horse."[23] However, like many of the Hundred Schools of Thought during the Warring States period (403–221 BC), the doctrine of the Mohist sect had little impact on the course of later Chinese thought, while this passage and others from the Mojing were only given serious attention by modern scholarship after the work of Joseph Needham in 1962.[23]
Aware of underground minerals associated with certain plants by at least the 5th century BC, the Chinese extracted trace elements of copper from Oxalis corniculata, pictured here, as written in the 1421 text Precious Secrets of the Realm of the King of Xin.
  • Geobotanical prospecting: Geobotanical prospecting can be defined as the connection made between the types of vegetation that grow in certain areas and the minerals that can be found underground in those same areas; this observation was first made in China.[27] It is now established in modern geobotany that only certain plants can grow in soils which are rich in certain types of minerals, such as Viola calaminaria and Thlaspi which grow in soils rich in zinc.[27] The Zhou Dynasty (c. 1050–256 BC) Chinese Classic of Mountains and Rivers, compiled from the 6th to 2nd centuries BC, states that a certain "huitang" plant only grows near ore deposits of gold.[27] As seen in the 5th century BC text Tribute of Yu, geobotanical prospecting in ancient China was mainly concerned with describing the nature of soil in different regions for agricultural purposes.[27] The Book of Master Wen, compiled by 380 AD and containing material from as far back as the 3rd century BC, states that the branches of trees tend to droop in soils where an abundance of jade is to be found.[28] In about 290 AD, Zhang Hua (232–300) wrote that hematite was found in abundance in any soil where smartweed grew.[28] In the Illustrated Mirror of the Earth, written in the early 6th century AD, there is a description of a plant with an elegant yellow stalk which was found to grow above copper, and another description of a plant with green leaves and a red stalk where lead is often found below.[28] In his Miscellaneous Morsels from Youyang, the Tang Dynasty (618–907) author Duan Chengshi (d. 863) noted that silver could often be found in the soil where ciboule onion grew, gold where a certain kind of shallot grew, and copper where ginger grew.[28] Su Song (1020–1101) of the Song Dynasty (960–1279) described how Portulaca oleracea could yield mercury if pounded, dried, and allowed to decay.[28] The Precious Secrets of the Realm of the King of Xin, written in 1421 during the Ming Dynasty (1368–1644), described how mineral trace elements were observed and could be extracted from certain plants, such as copper from Oxalis corniculata, gold from rape turnip, silver from weeping willows, and lead and tin from mugwort, chestnut, barley, and wheat.[28] Geobotanical prospecting was unknown in the rest of the world until about 1600 when Sir Thomas Challoner and his first cousin Thomas Challoner discovered alum mines on the former's property of Belman Bank, Guisborough, Yorkshire, England.[23] Both Challoner relatives realized here (and later in Italy) that leaves of oak trees were a much darker, richer green and their branches stronger and more spread out where the alum was to be found.[23]
  • Jia Xian triangle: This triangle was the same as Pascal's Triangle, discovered by Jia Xian in the first half of the 11th century, about six centuries before Pascal. Jia Xian used it as a tool for extracting square and cubic roots. The original book by Jia Xian titled Shi Suo Suan Shu was lost; however, Jia's method was expounded in detail by Yang Hui, who explicitly acknowledged his source: "My method of finding square and cubic roots was based on the Jia Xian method in Shi Suo Suan Shu."[30] A page from the Yongle Encyclopedia preserved this historic fact.
Mohandas Karamchand Gandhi tends to a leper; the Chinese were the first to describe the symptoms of leprosy.
Iron plate with an order 6 magic square in Arabic numbers from China, dating to the Yuan Dynasty (1271-1368).
  • Magic squares: The earliest magic square is the Lo Shu square, dating to 4th century BCE China. The square was viewed as mystical, and according the Chinese mythology, and "was first seen by Emperor Yu."[35]
With the description in Han Ying's written work of 135 BC (Han Dynasty), the Chinese were the first to observe that snowflakes had a hexagonal structure.
Oiled garments left in the tomb of Emperor Zhenzong of Song (r. 997–1022), pictured here in this portrait, caught fire seemingly at random, a case which a 13th-century author related back to the spontaneous combustion described by Zhang Hua (232–300) around 290 AD
  • Snowflake, observation of its hexagonal structure: In his Moral Discourses Illustrating the Han Text of the Book of Songs of 135 BC, the Han Dynasty (202 BC– 220 AD) author Han Ying wrote: "Flowers of plants and trees are generally five-pointed, but those of snow, which are called ying, are always six pointed."[48] This was the first explicit reference in world history to the hexagonal structure of snowflakes.[48] From then on, Chinese writers throughout the centuries mentioned the hexagonal structure of snowflakes, including the crown prince and poet Xiao Tong (501–531) and the Neo-Confucian philosopher Zhu Xi (1130–1200).[48] In contrast to Western ideas of snowflakes, Olaus Magnus (1490–1557) wrote in his A Description of the Northern Peoples in 1555 that snowflakes could take on many shapes, including crescents, arrows, nails, bells, and even the shape of the human hand.[23] It was not until 1591 that Thomas Hariot (1560–1621) recognized the snowflake's hexagonal structure, but he did not publish his jotted private notes on the subject.[23] Finally, the astronomer Johannes Kepler (1571–1630) wrote the first known European publication on the subject in 1611, the fifteen-page A New Year's Gift, or On the Six-Cornered Snowflake.[48]
  • Solar wind, observation of via comet tails: In the Book of Jin compiled during the Tang Dynasty (618–907), a passage written in 635 AD states: "In general, when a comet appears in the morning, its tail points towards the west, and when it appears in the evening, its tail points towards the east. This is a constant rule. If the comet is north or south of the Sun, its tail always points following the same direction as the light radiating from the Sun."[49] In other words, as Robert Temple states, "the Chinese observations of comet tails had been refined enough to establish the principle that comet tails always point away from the sun."[49] Furthermore, the text reveals that astronomers by at least the Tang Dynasty understood that, like the Moon, the light shining from a comet was merely reflected sunlight;[49] from the writings of Jing Fang (78–37 BC), Wang Chong (27–100), Zhang Heng (78–139), and others it is apparent that the Chinese already by the Han Dynasty (202 BC – 220 AD) understood that the Moon was illuminated solely by the Sun's rays of light.[50] Although the Chinese explained this constant rule about comets in terms of supernatural qi, it is now understood in modern astronomy as the concept of 'solar wind', where the powerful force of radiation from the Sun causes comets to turn away from it.[49]
  • Spontaneous combustion, recognition of: In his Record of Strange Things written sometime before 290 AD, the Jin Dynasty official and poet Zhang Hua (232–300) wrote the earliest known account acknowledging spontaneous combustion: "If ten thousand piculs of oil are accumulated in store, the oil will ignite itself spontaneously. The calamitous fire which occurred in the arsenal of the time of the Emperor Wu [of the Jin Dynasty] in the Taishi reign-period [265–74 AD] was caused by the stored oil."[51] There were other mentionings of spontaneous combustion in early Chinese literary works, while more often than not fires were blamed on arsonists.[52] The 13th-century work Parallel Cases Solved by Eminent Judges recounts an event in 1050 where imperial guards were charged in a court of law with the crime of allowing a fire to spread in the palace at Kaifeng; their sentence was commuted from the death penalty to a light punishment when artisans confessed that the chemical-enhanced (perhaps quicklime) oily curtains they made had the propensity to catch fire spontaneously when left out in the open, a statement which convinced Emperor Renzong (r. 1022–1063) since a random fire had recently started in oiled garments of Emperor Zhenzong's (r. 997–1022) mausoluem.[52] The author of Parallel Cases Solved by Eminent Judges noted that Zhang Hua had once believed oil stored in an arsenal spontaneously combusted, yet he concludes that what happened in that ancient arsenal was most likely the result of oiled garments, not just oil by itself.[52] The first acknowledgement of spontaneous combustion anywhere else in the world was made by J. P. F. Duhamel in a French scientific paper published in 1757, in which he described oiled canvas sails catching fire after being left out in the summer sun for only a few hours.[52]
  • Sunspots, recognition of as solar phenomena: The astronomer Gan De (fl. 4th century BC) from the State of Qi during the Warring States period (403–221 BC) was the first known writer to attribute sunspots as characteristics of the sun and true solar phenomena.[53] The next known recording of a sunspot in China was in 165 BC, yet the first precisely dated sunspot observed from China occurred on May 10, 28 BC, during the Han Dynasty (202 BC – 220 AD).[53] From 28 BC to 1368 AD, a total of 112 other instances of sunspots were recorded by the Chinese.[54] In the West, from the time of Aristotle (384–322 BC) of ancient Greece to the time of Galileo Galilei (1564–1642), it was commonly believed that the heavens were perfect, including the sun.[53] After the first written observation in the West of sunpots by Einhard (d. 840) in his Life of Charlemagne in 807 AD, the sun's periodic blemishes were explained by Western thinkers as being small invisible satellites or transits of Mercury and Venus; it was only in the 17th century that these beliefs were overturned.[55]
  • True north, concept of: The Song Dynasty (960–1279) official Shen Kuo (1031–1095), alongside his colleague Wei Pu, improved the orifice width of the sighting tube to make nightly accurate records of the paths of the moon, stars, and planets in the night sky, for a continuum of five years.[56] By doing so, Shen fixed the outdated position of the pole star, which had shifted over the centuries since the time Zu Geng (fl. 5th century) had plotted it; this was due to the precession of the Earth's rotational axis.[57][58] When making the first known experiments with a magnetic compass, Shen Kuo wrote that the needle always pointed slightly east rather than due south, an angle he measured which is now known as magnetic declination, and wrote that the compass needle in fact pointed towards the magnetic north pole instead of true north (indicated by the current pole star); this was a critical step in the history of accurate navigation with a compass.[59][60][61]


  • Culturing Chlamydia trachomatis bacteria: Chlamydia trachomatis agent was first cultured in the yolk sacs of eggs by Chinese scientists in 1957 [68]
  • Feathered theropods: The first feathered dinosaur outside of Avialae, Sinosauropteryx, meaning "Chinese reptilian wing," was discovered in the Yixian Formation by Chinese paleontologists in 1996.[69] The discovery is seen as evidence that dinosaurs originated from birds, a theory proposed and supported decades earlier by paleontologists like Gerhard Heilmann and John Ostrom, but "no true dinosaur had been found exhibiting down or feathers until the Chinese specimen came to light."[70] The dinosaur was covered in what are dubbed 'protofeathers' and considered to be homologous with the more advanced feathers of birds,[71] although some scientists disagree with this assessment.[72]
  • Hua's identity: In algebra, Hua's identity[73] states that for any elements a, b in a division ring, :a - (a^{-1} + (b^{-1} - a)^{-1})^{-1} = aba whenever ab \ne 0, 1. Replacing b with -b^{-1} gives another equivalent form of the identity: :(a+ab^{-1}a)^{-1} + (a+b)^{-1} =a^{-1}.
  • Heterosis in rice, three-line hybrid rice system: A team of agricultural scientists headed by Yuan Longping applied heterosis to rice, developing the three-line hybrid rice system in 1973.[75] The innovation allowed for roughly 12,000 kg (26,450 lbs) of rice to be grown per hectare (10,000 m2). Hybrid rice has proven to be greatly beneficial in areas where there is little arable land, and has been adopted by several Asian and African countries. Yuan won the 2004 Wolf Prize in agriculture for his work.[76]
  • Ky Fan norms: The sum of the k largest singular values of M is a matrix norm, the Ky Fan k-norm of M.The first of the Ky Fan norms, the Ky Fan 1-norm is the same as the operator norm of M as a linear operator with respect to the Euclidean norms of Km and Kn. In other words, the Ky Fan 1-norm is the operator norm induced by the standard l2 Euclidean inner product.

See also[edit]


  1. ^ Chern later acquired American citizenship in 1961. He was born in Jiaxing, Zhejiang.
  2. ^ Yang later acquired American citizenship in 1964, Lee in 1962. Both men were born in China.



  1. ^ a b c d Ho (1991), 516.
  2. ^ Temple (1986), 139.
  3. ^ Temple (1986), 142–143.
  4. ^ a b c d e Temple (1986), 143.
  5. ^ a b Needham (1986), Volume 3, 24–25.
  6. ^ Straffin (1998), 165.
  7. ^ a b c d e Temple (1986), 131.
  8. ^ a b Temple (1986), 132.
  9. ^ Medvei (1993), 49.
  10. ^ a b c Temple (1986), 133.
  11. ^ Temple (1986), 128–129.
  12. ^ Temple (1986), 127.
  13. ^ Temple (1986), 130.
  14. ^ Temple (1986), 199.
  15. ^ McClain and Ming (1979), 206.
  16. ^ McClain and Ming (1979), 207–208.
  17. ^ McClain and Ming (1979), 212.
  18. ^ Needham (1986), Volume 4, Part 1, 218–219.
  19. ^ Kuttner (1975), 166–168.
  20. ^ Needham (1986), Volume 4, Part 1, 227–228.
  21. ^ a b Temple (1986), 209.
  22. ^ a b Needham (1986), Volume 4, Part 1, 223.
  23. ^ a b c d e f Temple (1986), 161.
  24. ^ Needham (1986), Volume 3, 24–25, 121.
  25. ^ Shen, Crossley, and Lun (1999), 388.
  26. ^ Straffin (1998), 166.
  27. ^ a b c d Temple (1986), 159.
  28. ^ a b c d e f Temple (1986), 160.
  29. ^ a b c Temple (1986), 142.
  30. ^ Wu Wenjun chief ed, The Grand Series of History of Chinese Mathematics Vol 5 Part 2, chapter 1, Jia Xian
  31. ^ a b c McLeod & Yates (1981), 152–153 & footnote 147.
  32. ^ Aufderheide et al, (1998), 148.
  33. ^ Salomon (1998), 12–13.
  34. ^
  35. ^ C. J. Colbourn; Jeffrey H. Dinitz (2 November 2006). Handbook of Combinatorial Designs. CRC Press. p. 525. ISBN 978-1-58488-506-1. 
  36. ^ a b c d Temple (1986), 141.
  37. ^ Teresi (2002), 65–66.
  38. ^ Neehdam (1986), Volume 3, 99–100.
  39. ^ a b Berggren, Borwein & Borwein (2004), 27
  40. ^ Arndt and Haenel (2001), 177
  41. ^ Wilson (2001), 16.
  42. ^ Needham (1986), Volume 3, 100–101.
  43. ^ Berggren, Borwein & Borwein (2004), 24–26.
  44. ^ Berggren, Borwein & Borwein (2004), 26.
  45. ^ Berggren, Borwein & Borwein (2004), 20.
  46. ^ Gupta (1975), B45–B48
  47. ^ Berggren, Borwein, & Borwein (2004), 24.
  48. ^ a b c d Temple (1986), 162.
  49. ^ a b c d Temple (1986), 34.
  50. ^ Needham (1986), Volume 3, 227 & 411–414.
  51. ^ Temple (1986), 166–167.
  52. ^ a b c d Temple (1986), 167.
  53. ^ a b c Temple (1986), 29.
  54. ^ Temple (1986), 30.
  55. ^ Temple (1986), 29–30.
  56. ^ Sivin (1995), III, 17–18.
  57. ^ Sivin (1995), III, 22.
  58. ^ Needham (1986), Volume 3, 278.
  59. ^ Sivin (1995), III, 21–22.
  60. ^ Elisseeff (2000), 296.
  61. ^ Hsu (1988), 102.
  62. ^ Chen, J.R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao 17: 385–386. 
  63. ^ Chen, J.R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica 16: 157–176. 
  64. ^ Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao 17: 385–386.
  65. ^ Cheng, Shiu Yuen (1975a). "Eigenfunctions and eigenvalues of Laplacian". Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2. Providence, R.I.: American Mathematical Society. pp. 185–193. MR 0378003. 
  66. ^ Chavel, Isaac (1984). Eigenvalues in Riemannian geometry. Pure Appl. Math. 115. Academic Press. 
  67. ^ Chern, S. S. (1946). "Characteristic classes of Hermitian Manifolds". Annals of Mathematics. Second Series (The Annals of Mathematics, Vol. 47, No. 1) 47 (1): 85–121. doi:10.2307/1969037. ISSN 0003-486X. JSTOR 1969037. 
  68. ^ S Darougar, B R Jones, J R Kimptin, J D Vaughan-Jackson, and E M Dunlop. Chlamydial infection. Advances in the diagnostic isolation of Chlamydia, including TRIC agent, from the eye, genital tract, and rectum. Br J Vener Dis. 1972 December; 48(6): 416–420; TANG FF, HUANG YT, CHANG HL, WONG KC. Further studies on the isolation of the trachoma virus. Acta Virol. 1958 Jul-Sep;2(3):164-70; TANG FF, CHANG HL, HUANG YT, WANG KC. Studies on the etiology of trachoma with special reference to isolation of the virus in chick embryo. Chin Med J. 1957 Jun;75(6):429-47; TANG FF, HUANG YT, CHANG HL, WONG KC. Isolation of trachoma virus in chick embryo. J Hyg Epidemiol Microbiol Immunol. 1957;1(2):109-20
  69. ^ Ji Qiang; Ji Shu-an (1996). "On the discovery of the earliest bird fossil in China and the origin of birds" (PDF). Chinese Geology 233: 30–33. 
  70. ^ Browne, M.W. (19 October 1996). "Feathery Fossil Hints Dinosaur-Bird Link". New York Times. p. Section 1 page 1 of the New York edition. 
  71. ^ Chen Pei-ji, Pei-ji; Dong Zhiming; Zhen Shuo-nan. (1998). "An exceptionally preserved theropod dinosaur from the Yixian Formation of China". Nature 391 (6663): 147–152. Bibcode:1998Natur.391..147C. doi:10.1038/34356. 
  72. ^ Sanderson, K. (23 May 2007). "Bald dino casts doubt on feather theory". doi:10.1038/news070521-6. Retrieved 14 January 2011. 
  73. ^ Cohn 2003, §9.1
  74. ^ Hua Loo-keng (1938). "On Waring's problem". Quarterly Journal of Mathematics 9 (1): 199–202. doi:10.1093/qmath/os-9.1.199. 
  75. ^ Sant S. Virmani, C. X. Mao, B. Hardy, (2003). Hybrid Rice for Food Security, Poverty Alleviation, and Environmental Protection. International Rice Research Institute. ISBN 971-22-0188-0, p. 248
  76. ^ Wolf Foundation Agricultural Prizes
  77. ^ Huang-Minlon Journal of the American Chemical Society 1946, 68, 2487.
  78. ^ Huang-Minlon Journal of the American Chemical Society 1949, 71, 3301.
  79. ^ Organic Syntheses, Coll. Vol. 4, p. 510 (1963); Vol. 38, p. 34 (1958). (Article)
  80. ^ Yang, C. N.; Lee, T. D. (1952). "Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation". Physical Review 87: 404–409. Bibcode:1952PhRv...87..404Y. doi:10.1103/PhysRev.87.404. ISSN 0031-9007. 
  81. ^ Tsen, C. (1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper". J. Chinese Math. Soc. 171: 81–92. Zbl 0015.38803. 
  82. ^ Wu, Wen-Tsun (1978). "On the decision problem and the mechanization of theorem proving in elementary geometry". Scientia Sinica 21. 
  83. ^ P. Aubry, D. Lazard, M. Moreno Maza (1999). On the theories of triangular sets. Journal of Symbolic Computation, 28(1–2):105–124


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