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| '''Mathematics in China''' emerged independently by the 11th century BC.<ref>[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Chinese_overview.html Chinese overview<!-- Bot generated title -->]</ref> The Chinese independently developed very large and [[negative number]]s, [[decimal]]s, a place value decimal system, a [[Binary numeral system|binary system]], [[algebra]], [[geometry]], and [[trigonometry]]. Knowledge of Chinese [[mathematics]] before 254 BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. Dates centuries before the classical period are generally considered conjectural by Chinese scholars unless accompanied by verified archaeological evidence, in a direct analogue with the situation in the Far West. Neither Western nor Chinese archaeological findings comparable to those for [[history of mathematics|Babylonia or Egypt]] are known.
| | == 殺意があります == |
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| As in other early societies the focus was on [[astronomy]] in order to perfect the agricultural [[calendar]], and other practical tasks, and not on establishing [[formal systems]]. Ancient Chinese mathematicians did not develop an axiomatic approach, but made advances in algorithm development and algebra. The algorithm and algebra tradition of ancient Chinese together with the axiomic deduction of Greece formed the two equally important pillars of world mathematics. While the Greek mathematics declined in the west during the mediaeval times, the achievement of Chinese algebra reached its zenith in the 13th century, when [[Zhu Shijie]] invented method of four unknowns.
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| As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and that of the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when ''[[The Nine Chapters on the Mathematical Art]]'' reached its final form, while the ''[[Writings on Reckoning]]'' and ''[[Huainanzi]]'' are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or [[number theory]]. The [[Pythagorean_theorem#History|Pythagorean theorem]] for example, [[Zhou Bi Suan Jing|has been attested]] to the time of the [[Duke of Zhou]]. Knowledge of [[Pascal's triangle]] has also been shown to have existed in China centuries before [[Blaise Pascal|Pascal]],<ref>[http://www.psupress.psu.edu/books/titles/0-271-01238-2.html Frank J. Swetz and T. I. Kao: Was Pythagoras Chinese?<!-- Bot generated title -->]</ref> such as by [[Shen Kuo]].
| | == ' この名前、心あなたが怖がって聞いたが、 == |
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| == Early Chinese mathematics ==
| | 散乱始めているGangqi。<br>「よく、彼は最高のMogong木綿で大切除！、大削り取る手術クラス超自然をキャストしたい！カットするようにしてください、そうでないと思ったが、永遠の破壊をカットする必要があります！ [http://www.aseanacity.com/webalizer/prada-bags-30.html プラダ 財布] '<br>ヤン区日数黒ギザギザ段ボール道路から出てくる水を見悪魔、助けることが驚きませんでした [http://www.aseanacity.com/webalizer/prada-bags-33.html プラダ メンズ ベルト]。<br>'大きな切除 [http://www.aseanacity.com/webalizer/prada-bags-27.html プラダ 財布 値段]！' この名前、心あなたが怖がって聞いたが、<br>サイド寒さが、残余の魂をクリーンアップする悪魔未満のマスター木綿セーバーエクセルので、それは本当に垂直方向と水平方向の世界であることができ、薄い死んだラクダ馬よりも大きいとは思いませんでした。<br>「Tiens布ギャング！ [http://www.aseanacity.com/webalizer/prada-bags-33.html 財布 プラダ レディース] '<br>ためらい、正方形の冷たい心の動きは、数フィートの厚Gangqi壁への前日の突然3獅子は自分の目の前に抵抗する。<br>'カット [http://www.aseanacity.com/webalizer/prada-bags-25.html prada トートバッグ]！' 牙漢江におけるhuanhangrnはちょうどこのアクションを完了し、水の狂気の悪魔ビッグカット |
| [[File:Chinese pythagoras.jpg|thumb|280px|Visual proof for the (3, 4, 5) triangle as in the [[Zhou Bi Suan Jing]] 500–200 BC.]] | | 相关的主题文章： |
| [[File:Oracle numeral.jpg|thumb|right|280px|Oracle bone script decimal]]
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| [[File:Chounumerals.jpg|thumb|right|280px|counting rod place value decimal]]
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| Simple mathematics on [[Oracle bone script]] date back to the [[Shang Dynasty]] (1600–1050 BC). One of the oldest surviving mathematical works is the ''[[I Ching|Yi Jing]]'', which greatly influenced written literature during the [[Zhou Dynasty]] (1050–256 BC). For mathematics, the book included a sophisticated use of [[hexagram]]s. Leibniz pointed out, the I Ching contained elements of
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| Since the Shang period, the Chinese had already fully developed a [[decimal]] system. Since early times, Chinese understood basic [[arithmetic]] (which dominated far eastern history), [[algebra]], [[equations]], and [[negative numbers]] with [[counting rods]].{{Citation needed|date=October 2008}} Although the Chinese were more focused on arithmetic and advanced algebra for [[astronomy|astronomical]] uses, they were also the first to develop negative numbers, [[algebraic geometry]] (only Chinese geometry) and the usage of decimals.
| | == to escape out == |
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| Math was one of the ''Liù Yì'' (六艺) or ''[[Six Arts]]'', students were required to master during the [[Zhou Dynasty]] (1122–256 BC). Learning them all perfectly was required to be a perfect gentleman, or in the Chinese sense, a "[[Polymath|Renaissance Man]]". Six Arts have their roots in the [[Confucian philosophy]].
| | Energy shot for me, but I do not care, I'll let you [http://www.aseanacity.com/webalizer/prada-bags-34.html prada] see that I am [http://www.aseanacity.com/webalizer/prada-bags-27.html プラダ 財布 スタッズ] more powerful than the strength of the wind edge, so you know do not disgrace my servant but a glory. '<br>heart Huang Son look at all this is very [http://www.aseanacity.com/webalizer/prada-bags-25.html プラダ人気財布] accurate, dòng police people.<br>'Then you try.' bare Senior Gu Changfeng not able to tolerate kneeling here, everything is not able to tolerate it, can not wait to be broken seal, to escape out, [http://www.aseanacity.com/webalizer/prada-bags-22.html プラダ 財布 定価] crazy vent, then penance supernatural [http://www.aseanacity.com/webalizer/prada-bags-34.html prada] powers, looking for party cold revenge.<br>'Well!'<br>heart Wong Son face sè look turned serious: 'You ready for my magic will directly damage the seal, if you could not stand, it will blown powder.' between<br>speak, one of his big hands in the air immediately appeared many free runes, these runes formed a huge statue of the heart, like lotus tires in general, that huge heart center |
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| The oldest existent work on geometry in China comes from the philosophical Mohist canon of c. 330 BC, compiled by the followers of [[Mozi]] (470–390 BC). The ''Mo Jing'' described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point.<ref name="needham volume 3 91">Needham, Volume 3, 91.</ref> Much like [[Euclid]]'s first and third definitions and [[Plato]]'s 'beginning of a line', the ''Mo Jing'' stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it."<ref name="needham volume 3 92">Needham, Volume 3, 92.</ref> Similar to the [[atomist]]s of [[Democritus]], the ''Mo Jing'' stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved.<ref name="needham volume 3 92"/> It stated that two lines of equal length will always finish at the same place,<ref name="needham volume 3 92"/> while providing definitions for the ''comparison of lengths'' and for ''parallels'',<ref name="needham volume 3 92 93">Needham, Volume 3, 92-93.</ref> along with principles of space and bounded space.<ref name="needham volume 3 93">Needham, Volume 3, 93.</ref> It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.<ref name="needham volume 3 93 94">Needham, Volume 3, 93-94.</ref> The book provided word recognition for circumference, diameter, and radius, along with the definition of volume.<ref name="needham volume 3 94">Needham, Volume 3, 94.</ref>
| | == Grip to the unseen world == |
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| The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the ''[[Zhou Bi Suan Jing]]'' dates around 1200–1000 BC, yet many scholars believed it was written between 300–250 BC. The ''Zhou Bi Suan Jing'' contains an in-depth proof of the ''Gougu Theorem '' ([[Pythagorean Theorem]]) but focuses more on astronomical calculations.
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| == Qin mathematics == | | == then there are countless lives Great concubine == |
| Not much is known about [[Qin dynasty]] mathematics, or before, due to the [[burning of books and burying of scholars]].
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| Knowledge of this period must be carefully determined by their civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were incredible feats of human engineering. Emperor [[Qin Shihuang]]（秦始皇）ordered many men to build large, lifesize statues for the palace tomb along with various other temples and shrines. The shape of the tomb is designed with geometric skills of architecture. It is certain that one of the greatest feats of human history; the great wall required many mathematical "techniques." All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion.
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| ==Han mathematics== | | == if this can be the eldest of all ages == |
| {{Further|Science and technology of the Han Dynasty#Mathematics and astronomy}}
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| [[File:九章算術.gif|thumb|''[[The Nine Chapters on the Mathematical Art]]''.]]
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| In the Han Dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of [[counting rods]] called [[Rod calculus|chousuan]],consisted of only nine symbols, a blank space on the counting board stood for zero. The mathematicians [[Liu Xin]] (d. 23) and [[Zhang Heng]] (78–139) gave more accurate approximations for [[pi]] than Chinese of previous centuries had used. Zhang also applied mathematics in his [[Chinese astronomy|work in astronomy]].
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| ===Suan shu shu===
| | Han beard suddenly stop down, breathing heavily.<br>'Well, here we are [http://www.aseanacity.com/webalizer/prada-bags-27.html プラダ最新財布] refining today, anyway, this is almost full power of magic, Ula fellow, it seems your position in points among the firm Baoyan, what material there is nothing material to be even Once upon a time even began some are willing to allocate it, how it hundreds of years among fellow immortal also practicing [http://www.aseanacity.com/webalizer/prada-bags-21.html prada ピンク 財布] to the peak, if this can be the eldest of all ages, space law is certain to be able to practice , it is asking for the moon. '<br>good woman can stay down, shut the fire, 'I have to go back and rest.'<br>'I have to go back.' mysterious old man said.<br>'Well, I [http://www.aseanacity.com/webalizer/prada-bags-24.html プラダ 財布 値段] stayed to see Shoudan furnace, lest other Lian [http://www.aseanacity.com/webalizer/prada-bags-29.html prada メンズ 財布] Po division to peep.' sat down cross-legged on the cold side, [http://www.aseanacity.com/webalizer/prada-bags-34.html プラダ 長財布] silently Breath, 'Now this magic wand, that is temperature dependent, waiting for Missy comes, we the furnace together, came up with the world |
| The ''[[Suàn shù shū]]'' (writings on reckoning) is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1984 when [[archaeologist]]s opened a tomb at Zhangjiashan in [[Hubei]] province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western [[Han dynasty]]. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the ''Suan shu shu'' is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. Some linguistic hints point back to the [[Qin dynasty]].
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| In an example of an elementary mathematics in the ''Suàn shù shū'', the [[square root]] is approximated by using an "excess and deficiency" method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."<ref>Dauben, p 210.</ref>
| | == their saver again pulled Abi gas == |
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| ===The Nine Chapters on the Mathematical Art===
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| ''[[The Nine Chapters on the Mathematical Art]]'' is a Chinese [[mathematics]] book, its oldest archeological date being 179 AD (traditionally dated 1000 BC), but perhaps as early as 300–200 BC. Although the author(s) are unknown, they made a huge contribution in the eastern world. The methods were made for everyday life and gradually taught advanced methods. It also contains evidence of the [[Gaussian elimination]] and [[Cramer's Rule]] for [[rod calculus#System of linear equations|system of linear equations]]. | | 相关的主题文章： |
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| {{confusing|section|date=April 2012}}
| | == he is jealous of pity == |
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| It was one of the most influential of all Chinese mathematical books and it is composed of some 246 problems. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns.<ref>Boyer, 1991, "Chinese Math, China and India"</ref> Estimates concerning the Chou Pei Suan Ching, generally considered to be the oldest of the mathematical classics, differ by almost a thousand years. A date of about 300 BC would appear reasonable, thus placing it in close competition with another treatise, the Jiu zhang suanshu, composed about 250 BC, that is, shortly before the Han dynasty (202 BC). Almost as old at the Chou Pei, and perhaps the most influential of all Chinese mathematical books, was the [[Jiuzhang suanshu]], or Nine Chapters on the Mathematical Art. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. Chapter eight of the Nine chapters is significant for its solution of problems of simultaneous linear equations, using both positive and negative numbers. The earliest known [[magic squares]] appeared in China.<ref>Boyer, 1991, "Magic Square, China and India"</ref> The Chinese were especially fond of patterns, as a natural outcome of arranging [[counting rods]] in rows on counting board to carry out computation; hence,it is not surprising that the first record (of ancient but unknown origin) of a magic square appeared there. The concern for such patterns led the author of the ''Nine Chapters'' to solve the [[Rod calculus#System of linear equations|system of simultaneous linear equations]] by placing the coefficients and constant terms of the linear equations into a matrix and performing column reducing operations on the matrix to reduce it to a triangular form represented by the equations 36z = 99, 5y + z = 24, and 3x + 2y + z = 39 from which the values of z, y, and x are successively found with ease. The last problem in the chapter involves four equations in five unknowns, and the topic of [[indeterminate equations]] was to remain a favorite among Oriental peoples.
| | There are a lot of treasure, Wang Pin cents. Once looted, the [http://www.aseanacity.com/webalizer/prada-bags-20.html prada リボン 財布] consequences could be disastrous, and there I [http://www.aseanacity.com/webalizer/prada-bags-33.html 財布 プラダ] am generally not able to come in! And heaven only ancient imperial many people to discuss, the meeting together, or Heaven's Soldiers behest, before they can enter. '<br>'hateful ah damn! so quickly put this child in front of eighteen layers looted, valuable things all taken away.'<br>disaster Wong is also angry rampage three dead [http://www.aseanacity.com/webalizer/prada-bags-23.html prada 財布 新作] god, he is jealous of [http://www.aseanacity.com/webalizer/prada-bags-27.html プラダ最新財布] pity, what a rich treasure of heaven, and it was all looted or [http://www.aseanacity.com/webalizer/prada-bags-35.html プラダ 財布 レディース] plundered an enemy, and no one will not be easy.<br>and disaster Huang usually not able to enter heaven treasure, and that he not get the baby.<br>Huangpu Avenue This is the first time to enter heaven treasure, envy, heaven is heaven than the treasure trove of Divine Pure Land is much greater, he even wants to plunder something, but no, no guts.<br>however, saw the cold side, a large fishing |
| | | 相关的主题文章： |
| ==Mathematics in the period of disunity==
| | <ul> |
| [[File:Sea island survey.jpg|thumb|right|100px|Lui Hui's Survey of sea island]]
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| [[File:Sunzi division.GIF|thumb|left|100px|Sunzi algorithm for division 400 AD]]
| | <li>[http://baotouhuaqi68.bgp103.kaiyele.com/home/space.php?uid=92055&do=blog&id=431599 http://baotouhuaqi68.bgp103.kaiyele.com/home/space.php?uid=92055&do=blog&id=431599]</li> |
| [[File:AL Khwarizmi division.GIF|right|thumb|100px|al Khwarizmi division in 9th century]]
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| In the third century [[Liu Hui]] wrote his commentary on the Nine Chapters and also wrote [[Haidao suanjing]] which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium.<ref>Frank J. Swetz: The Sea Island Mathematical Manual, Surveying and Mathematics in Ancient China 4.2 Chinese Surveying Accomplishments, A Comparative Retrospection p63 The Pennsylvania State University Press, 1992 ISBN 0-271-00799-0</ref> He was the first Chinese mathematician to calculate ''π''=3.1416 with his [[Liu Hui's π algorithm|''π'' algorithm]]. He discovered the usage of [[Cavalieri's principle]] to find an accurate formula for the volume of a cylinder, and also developed elements of the [[integral calculus|integral]] and the [[differential calculus|differential]] [[calculus]] during the 3rd century CE.
| | <li>[http://www.liuguohuan.com/plus/feedback.php?aid=16 http://www.liuguohuan.com/plus/feedback.php?aid=16]</li> |
| [[File:Diaorifa.GIF|thumb|right|100px|fraction interpolation for pi]]
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| In the fourth century, another influential mathematician named [[Zu Chongzhi]], introduced the ''Da Ming Li.'' This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little is really known about his life. Today, the only sources are found in [[Book of Sui]], we now know that Zu Chongzhi was one of the generations of mathematicians. He used Liu Hui's pi-algorithm applied to a 12288-gon and obtained a value of pi to 7 accurate decimal places (between 3.1415926 and 3.1415927), which would remain the most accurate approximation of π available for the next 900 years. He also used He Chengtian's interpolation method for approximating irrational number with fraction in his astronomy and mathematical works, he obtained<math>\tfrac{355}{113}</math> as a good fraction approximate for pi; Yoshio Mikami commented that neither the Greeks, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585, "the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe"<ref>[[Yoshio Mikami]], The Development of Mathematics in China and Japan, chap 7, p. 50, reprint of 1913 edition Chelsea, NY, Library of Congress catalog 61–13497</ref> Along with his son, Zu Geng, Zu Chongzhi used the Cavalieri Method to find an accurate solution for calculating the volume of the sphere. His work, ''Zhui Shu'' was discarded out of the syllabus of mathematics during the Song dynasty and lost. Many believed that ''Zhui Shu'' contains the formulas and methods for [[Linear algebra|linear]], [[Matrix (mathematics)|matrix algebra]], algorithm for calculating the value of ''π'', formula for the volume of the sphere. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics.
| | <li>[http://www.topledtube.com/plus/feedback.php?aid=140 http://www.topledtube.com/plus/feedback.php?aid=140]</li> |
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| A mathematical manual called "Sunzi mathematical classic" dated around 400 CE contained the most detailed step by step description of [[Rod calculus#Multiplication|multiplication]] and division algorithm with counting rods. The earliest record of multiplication and division algorithm using [[Hindu Arabic numerals]] was in writing by [[Al Khwarizmi]] in early 9th century. Khwarizmi's step by step division algorithm was completely identical to [[Rod calculus#Division|Sunzi division algorithm]] described in Sunzi mathematical classic four centuries earlier.<ref>Lam Lay Yong, The Development of Hindu Arabic and Traditional Chinese Arithematic, Chinese Science, 13(1996) 35–54{{sic?}}</ref> Khwarizmi's work was translated into Latin in the 13th century and spread to the west, the division algorithm later evolved into [[Galley division]]. The route of transmission of Chinese place value decimal arithmetic know how to the west is unclear, how Sunzi's division and multiplication algorithm with rod calculus ended up in Hindu Arabic numeral form in Khwarizmi's work is unclear, as al Khwarizmi never given any Sankrit source nor quoted any Sanskrit stanza. However, the influence of rod calculus on Hindu division is evident, for example in the division example, 324 should be 32400, only rod calculus used blanks for zeros.<ref>Lam Lay Yong, The Development of Hindu Arabic and Traditional Chinese Arithematics [http://sciences.aum.edu/~sbrown/Hindu%20Arabic%20and%20Chinese.pdf]</ref>
| | </ul> |
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| In the fifth century the manual called "Zhang Qiujian suanjing" discussed linear and quadratic equations. By this point the Chinese had the concept of [[negative numbers]].
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| ==Tang mathematics==
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| By the [[Tang Dynasty]] study of mathematics was fairly standard in the great schools. [[The Ten Computational Canons]] was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (李淳风 602－670），as the official mathematical texts for imperial examinations in mathematics.
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| [[Wang Xiaotong]] was a great mathematician in the beginning of the [[Tang Dynasty]], and he wrote a book: [[Jigu Suanjing]] (''Continuation of Ancient Mathematics''), in which cubic equations
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| appear for the first time<ref>Yoshio Mikami, Mathematics in China and Japan,p53</ref>
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| The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of [[Namri Songtsen|Nam-ri srong btsan]], who died in 630.<ref>{{cite book|url=http://books.google.com/books?id=XVHOAAAAMAAJ&pg=PA5826&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q=sixth%20century%20the%20tibetans%20obtained%20their%20first%20knowledge%20of%20arithmetic%20and%20medicine%20from%20the%20chinese&f=false|title=Americanized Encyclopædia Britannica: rev. and amended A dictionary of arts, sciences and literature, to which is added biographies of living subjects. 96 colored maps and numerous illustrations, Volume 9|author=|year=1890|publisher=Belford-Clarke co.|edition=|location=|page=5826|isbn=|accessdate=2011-07-01}}Americanized Encyclopædia Britannica: Rev. and Amended A Dictionary of Arts, Sciences and Literature, to which is Added Biographies of Living Subjects. 96 Colored Maps and Numerous Illustrations</ref><ref>{{cite book|url=http://books.google.com/books?id=-3saAAAAYAAJ&pg=PA5826&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC4Q6AEwAQ#v=onepage&q=sixth%20century%20the%20tibetans%20obtained%20their%20first%20knowledge%20of%20arithmetic%20and%20medicine%20from%20the%20chinese&f=false|title=The home encyclopædia: compiled and revised to date from the leading encyclopædias, Volume 18|author=|year=1895|publisher=Educational publishing co.|edition=|location=|page=5826|isbn=|accessdate=2011-07-01}}The Home Encyclopædia: Compiled and Revised to Date from the Leading Encyclopædias</ref><ref>{{cite book|url=http://books.google.com/books?id=Dj0hAQAAIAAJ&pg=PA5826&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDIQ6AEwAg#v=onepage&q=sixth%20century%20the%20tibetans%20obtained%20their%20first%20knowledge%20of%20arithmetic%20and%20medicine%20from%20the%20chinese&f=false|title=Americanized Encyclopædia Britannica, revised and amended: A dictionary of arts, sciences and literature; to which is added biographies of livings subjects ...|author=|year=1890|publisher=The "Examiner"|edition=|location=|page=5826|isbn=|accessdate=2011-07-01}}Volume 9 of Americanized Encyclopædia Britannica, Revised and Amended: A Dictionary of Arts, Sciences and Literature; to which is Added Biographies of Livings Subjects</ref><ref>{{cite book|url=http://books.google.com/books?id=uDQEAAAAYAAJ&pg=PA926&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDYQ6AEwAw#v=onepage&q=sixth%20century%20the%20tibetans%20obtained%20their%20first%20knowledge%20of%20arithmetic%20and%20medicine%20from%20the%20chinese&f=false|title=The encyclopædia britannica: a dictionary of arts, sciences, literature and general information, Volume 26|author=|editor=Hugh Chisholm|year=1911|publisher=At the University press|edition=11|location=|page=926|isbn=|accessdate=2011-07-01}}The Encyclopædia Britannica: A Dictionary of Arts, Sciences, Literature and General Information, Hugh Chisholm</ref><ref>{{cite book|url=http://books.google.com/books?id=ezZKAAAAYAAJ&pg=PA345&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDoQ6AEwBA#v=onepage&q=sixth%20century%20the%20tibetans%20obtained%20their%20first%20knowledge%20of%20arithmetic%20and%20medicine%20from%20the%20chinese&f=false|title=The Encyclopædia Britannica: a dictionary of arts, sciences, and general literature, Volume 23|author=|editor=Thomas Spencer Baynes|year=1888|publisher=C. Scribner's sons|edition=9|location=|page=345|isbn=|accessdate=2011-07-01}}The Encyclopædia Britannica: A Dictionary of Arts, Sciences, and General Literature, Thomas Spencer Baynes</ref><ref>{{cite book|url=http://books.google.com/books?id=gJlvzd4MzSUC&pg=PA926&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CD4Q6AEwBQ#v=onepage&q=sixth%20century%20the%20tibetans%20obtained%20their%20first%20knowledge%20of%20arithmetic%20and%20medicine%20from%20the%20chinese&f=false|title=The Encyclopædia Britannica: a dictionary of arts, sciences, literature and general information, Volume 26|author=Hugh Chisholm|year=1911|publisher=The Encyclopædia Britannica Co.|edition=11|location=|page=926|isbn=|accessdate=2011-07-01}}The Encyclopædia Britannica: A Dictionary of Arts, Sciences, Literature and General Information, Hugh Chisholm</ref><ref>{{cite book|url=http://books.google.com/books?id=QF5JAAAAYAAJ&pg=PA345&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEMQ6AEwBg#v=onepage&q=sixth%20century%20the%20tibetans%20obtained%20their%20first%20knowledge%20of%20arithmetic%20and%20medicine%20from%20the%20chinese&f=false|title=The Encyclopædia britannica: a dictionary of arts, sciences, and general literature ; the R.S. Peale reprint, with new maps and original American articles, Volume 23|author=William Harrison De Puy|year=1893|publisher=Werner Co.|edition=9|location=|page=345|isbn=|accessdate=2011-07-01}}The Encyclopædia Britannica: A Dictionary of Arts, Sciences, and General Literature ; the R.S. Peale Reprint, with New Maps and Original American Articles, William Harrison De Puy</ref><ref>{{cite book|url=http://books.google.com/books?id=rVXYAAAAMAAJ&pg=PA211&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=s-sNTpuECaL30gHl5fWtDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwADgK#v=onepage&q&f=false|title=The Life of the Buddha and the early history of his order: derived from Tibetan works in the Bkah-hgyur and Bstan-hgyur followed by notices on the early history of Tibet and Khoten|author=Translated by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio|year=1907|publisher=K. Paul, Trench, Trübner|edition=|location=|page=211|isbn=|accessdate=2011-07-01}}</ref><ref>{{cite book|url=http://books.google.com/books?id=59FAAAAAYAAJ&pg=PA211&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=s-sNTpuECaL30gHl5fWtDg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC0Q6AEwATgK#v=onepage&q&f=false|title=The life of the Buddha: and the early history of his order|author=William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio|year=1884|publisher=Trübner & co.|edition=|location=|page=211|isbn=|accessdate=2011-07-01}}</ref><ref>{{cite book|url=http://books.google.com/books?id=iIE9AAAAIAAJ&pg=PA211&dq=sixth+century+the+tibetans+obtained+their+first+knowledge+of+arithmetic+and+medicine+from+the+chinese&hl=en&ei=OOgNTsXSC6rx0gH_somqDg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFMQ6AEwCQ#v=onepage&q&f=false|title=The Life of the Biddha and the Early History of His Order Derived from Tibetan Works in the Bkah-hgyur and Bstan-khoten|author=|year=|publisher=Taylor & Francis|edition=|location=|page=211|isbn=|accessdate=2011-07-01}}</ref>
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| The [[Aryabhata's sine table|table]] of [[Trigonometric functions|sines]] by the [[Indian mathematics|Indian mathematician]], [[Aryabhata]], were translated into the Chinese mathematical book of the ''[[Treatise on Astrology of the Kaiyuan Era|Kaiyuan Zhanjing]]'', compiled in 718 AD during the Tang Dynasty.<ref name="needham volume 3 109">Needham, Volume 3, 109.</ref> Although the Chinese excelled in other fields of mathematics such as solid [[geometry]], [[binomial theorem]], and complex [[algebra]]ic formulas,early forms of [[trigonometry]] were not as widely appreciated as in the contemporary Indian and [[Islamic mathematics]].<ref name="needham volume 3 108 109">Needham, Volume 3, 108-109.</ref> I-Xing, the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an [[empirical]] substitute known as ''chong cha'', while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.<ref name="needham volume 3 109"/>
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| ==Song and Yuan mathematics==
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| [[Northern Song Dynasty]] mathematician [[Jia Xian]] developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule.<ref>Martzloff, 142</ref>
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| [[File:Yanghui triangle.gif|thumb|Yang Hui triangle ([[Pascal's triangle]]) using rod numerals, as depicted in a publication of [[Zhu Shijie]] in 1303 AD]]
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| Four outstanding mathematicians arose during the [[Song Dynasty]] and [[Yuan Dynasty]], particularly in the twelfth and thirteenth centuries: [[Yang Hui]], [[Qin Jiushao]], [[Li Zhi (mathematician)|Li Zhi]] (Li Ye), and [[Zhu Shijie]]. Yang Hui, Qin Jiushao, Zhu Shijie all used the [[Horner scheme|Horner]]-[[Ruffini's rule|Ruffini]] method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove "[[Pascal's Triangle]]", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD). Li Zhi on the other hand, investigated on a form of algebraic geometry based on [[Tian yuan shu]]. His book; [[Ceyuan haijing]] revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians.
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| Things grew quiet for a time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with [[Zhu Shijie]]'s two books ''[[Suanxue qimeng]]'' and the ''[[Siyuan yujian]]''. In one case he reportedly gave a method equivalent to [[Carl Friedrich Gauss|Gauss]]'s pivotal condensation.
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| [[Qin Jiushao]] (c. 1202–1261) was the first to introduce the [[0 (number)|zero symbol]] into Chinese mathematics.<ref>Needham, Volume 3, 43.</ref> Before this innovation, blank spaces were used instead of zeros in the system of [[counting rods]].<ref>Needham, Volume 3, 62–63.</ref> One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?"<ref>Yoshio Mikami, The development of Mathematics in China and Japan, p77 Leipzig, 1912</ref> Qin also solved a 10th order equation.<ref>Ulrich Librecht,Chinese Mathematics in the Thirteenth Century p. 211 Dover 1973</ref>
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| [[Pascal's triangle]] was first illustrated in China by Yang Hui in his book ''Xiangjie Jiuzhang Suanfa'' (详解九章算法), although it was described earlier around 1100 by [[Jia Xian]].<ref>Needham, Volume 3, 134–137.</ref> Although the ''Introduction to Computational Studies'' (算学启蒙) written by [[Zhu Shijie]] ([[floruit|fl.]] 13th century) in 1299 contained nothing new in Chinese [[algebra]], it had a great impact on the development of [[Japanese mathematics]].<ref>Needham, Volume 3, 46.</ref>
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| ===Algebra===
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| ====Ceyuan haijing====
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| {{main|Ceyuan haijing}}
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| [[File:圆城图式.jpg|thumb|right|200px|Li Ye's inscribed circle in triangle:'''Diagram of a round town''']]
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| [[File:Magic circle.jpg|thumb|right|200px|[[Yang Hui]]'s Magic Circle]]
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| [[Ceyuan haijing]] (pinyin: Cèyuán Hǎijìng) (Chinese characters:測圓海鏡), or ''Sea-Mirror of the Circle Measurements'', is a collection of 692 formula and 170 problems related to inscribed circle in a triangle, written by [[Li Zhi (mathematician)|Li Zhi]] (or Li Ye) (1192–1272 AD). He used [[Tian yuan shu]] to convert intricated geometry problems into pure algebra problems. He then used ''fan fa'', or [[Horner's method]], to solve equations of degree as high as six, although he did not describe his method of solving equations.<ref name="Boyer Sea Mirror">{{Harv|Boyer|1991|loc="China and India" p. 204}}</ref> "Li Chih (or Li Yeh, 1192–1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His ''Ts'e-yuan hai-ching'' (''Sea-Mirror of the Circle Measurements'') includes 170 problems dealing with[...]some of the problems leading to polynomial equations of sixth degree. Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (ca. 1202 – ca.1261) and Yang Hui (fl. ca. 1261–1275). "</ref>
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| ====''Jade Mirror of the Four Unknowns''====
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| {{cleanup|reason=<incoherent,did not touched upon the key significance of this treatise>|date=May 2012}}
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| [[File:Sixianghuiyuan.jpg|thumb|right|300px|Facsimile of Zhu Shijie's ''Jade Mirror of Four Unknowns'']] | |
| ''Si-yüan yü-jian''《四元玉鑒》, or ''Jade Mirror of the Four Unknowns'', was written by [[Zhu Shijie]] in 1303 AD and it marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. The ''Ssy-yüan yü-chien'' deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of ''fan fa'', today called [[Horner's method]], to solve these equations.{{Harv|Boyer|1991|loc="China and India" p. 203}} "The last and greatest of the Sung mathematicians was Chu Chih-chieh (fl. 1280–1303), yet we known little about him-, [...]Of greater historical and mathematical interest is the ''Ssy-yüan yü-chien''(''Precious Mirror of the Four Elements'') of 1303. In the eighteenth century this, too, disappeared in China, only to be rediscovered in the next century. The four elements, called ''heaven'', ''earth'', ''man'', and ''matter'', are the representations of four unknown quantities in the same equation. The book marks the peak in the development of Chinese algebra, for it deals with simultaneous equations and with equations of degrees as high as fourteen. In it the author describes a transformation method that he calls ''fan fa'', the elements of which to have arisen long before in China, but which generally bears the name of Horner, who lived half a millennium later."</ref>
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| The ''Jade Mirror'' opens with a diagram of the arithmetic triangle (Pascal's triangle) using a round zero symbol, but Chu Shih-chieh denies credit for it. A similar triangle appears in Yang Hui's work, but without the zero symbol.<ref name="boyer205">{{Harv|Boyer|1991|loc="China and India" p. 205}}</ref>
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| There are many summation series equations given without proof in the ''Precious mirror''. A few of the summation series are:<ref name="boyer205">{{Harv|Boyer|1991|loc="China and India" p. 205}} "A few of the many summations of series found in the ''Precious Mirror'' are the following:[...] However, no proofs are given, nor does the topic seem to have been continued again in China until about the nineteenth century. [...] The ''Precious Mirror'' opens with a diagram of the arithmetic triangle, inappropriately known in the West as "pascal's triangle." (See illustration.) [...] Chu disclaims credit for the triangle, referring to it as a "diagram of the old method for finding eighth and lower powers." A similar arrangement of coefficients through the sixth power had appeared in the work of Yang Hui, but without the round zero symbol."</ref>
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| :<math>1^2 + 2^2 + 3^2 + \cdots + n^2 = {n(n + 1)(2n + 1)\over 3!}</math>
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| :<math>1 + 8 + 30 + 80 + \cdots + {n^2(n + 1)(n + 2)\over 3!} = {n(n + 1)(n + 2)(n + 3)(4n + 1)\over 5!}</math>
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| :<math>6+48+180+</math>……<math>n^2(n+1)(n+2)=</math><math>1 \over 20</math><math>n(n+1)(n+2)(n+3)(4n+1)</math>
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| :<math>6+90+336+900+</math>……<math>n^2(n+1)(2n+1)=</math><math>1 \over 10</math><math>n(n+1)(n+2)(n(4n+1+1/2)+(4n+1/2))</math> | |
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| ====Mathematical Treatise in Nine Sections====
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| ''Shu-shu chiu-chang'', or [[Mathematical Treatise in Nine Sections]], was written by the wealthy governor and minister [[Ch'in Chiu-shao]] (ca. 1202 – ca. 1261 AD) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.<ref name="Boyer Sea Mirror" />
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| ====Magic Squares and Magic Circles====
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| The earliest known [[magic square]]s of order greater than three are attributed to [[Yang Hui]] (fl. ca. 1261–1275), who worked with magic squares of order as high as ten.<ref>{{Harv|Boyer|1991|loc="China and India" pp. 204–205}} "The same "Horner" device was used by Yang Hui, about whose life almost nothing is known and who work has survived only in part. Among his contributions that are extant are the earliest Chinese magic squares of order greater than three, including two each of orders four through eight and one each of orders nine and ten."</ref> He also worked with [[Magic circle (mathematics)|magic circle]].
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| ===Trigonometry===
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| The embryonic state of [[trigonometry]] in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations.<ref name="needham volume 3 109"/> The [[polymath]] Chinese scientist, mathematician and official [[Shen Kuo]] (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.<ref name="needham volume 3 109"/> Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc of a circle ''s'' by ''s'' = ''c'' + 2''v''<sup>2</sup>/''d'', where ''d'' is the [[diameter]], ''v'' is the [[versine]], ''c'' is the length of the chord ''c'' subtending the arc.<ref name="katz 308">Katz, 308.</ref> Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for [[spherical trigonometry]] developed in the 13th century by the mathematician and astronomer [[Guo Shoujing]] (1231–1316).<ref name="restivo 32">Restivo, 32.</ref> As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used [[spherical trigonometry]] in his calculations to improve the [[Chinese calendar|calendar system]] and [[Chinese astronomy]].<ref name="needham volume 3 109"/><ref name="gauchet 151">Gauchet, 151.</ref> Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:
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| <blockquote>
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| Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two [[meridian arc]]s, one of which passed through the [[summer solstice]] point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).<ref name="needham volume 3 109 110">Needham, Volume 3, 109–110.</ref>
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| </blockquote>
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| ==Later developments==
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| [[File:Boulier1.JPG|left|150px]]
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| However, after the overthrow of the [[Yuan Dynasty]] China became suspicious of knowledge it used. The [[Ming Dynasty]] turned away from math and physics in favor of [[botany]] and [[pharmacology]].
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| At this period, the [[abacus]] which first appeared in the second century BC<ref>{{Harvcolnb|Ifrah|2001|p=17}}</ref> now overtook the counting rods and became the preferred computing device. [[Zhu Zaiyu, Prince of Zheng]] who invented the [[equal temperament]] used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy.
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| Although this switch from counting rods to the abacus allowed for reduced computation times, it may have also led to the stagnation and decline of Chinese mathematics. The pattern rich layout of counting rod numerals on counting boards inspired many Chinese inventions in mathematics, such as the cross multiplication principle of fractions and methods for solving linear equations. Similarly, Japanese mathematicians were influenced by the counting rod numeral layout in their definition of the concept of a matrix. However, during the Ming dynasty, mathematicians were fascinated with perfecting algorithms for the abacus. As such, many works devoted to abacus mathematics appeared in this period; at the expense of new idea creation.
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| Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of ''[[Euclid's Elements]]'' by Chinese official and astronomer [[Xu Guangqi]] (1562–1633) and the Italian Jesuit [[Matteo Ricci]] (1552–1610).<ref name="needham volume 3 110">Needham, Volume 3, 110.</ref>
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| A revival of mathematics in China began in the late nineteenth century, when [[Joseph Edkins]], [[Alexander Wylie (missionary)|Alexander Wylie]] and [[Li Shanlan]] translated works on astronomy, algebra and differential-integral calculus into Chinese, published by London Missionary Press in Shanghai.
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| == Mathematical texts ==
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| '''Zhou Dynasty'''
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| ''Zhoubi Suanjing" c. 1000 BCE-100 CE
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| -Astronomical theories, and computation techniques
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| -Proof of the Pythagorean theorem (Shang Gao Theorem)
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| -Fractional computations
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| -Pythagorean theorem for astronomical purposes
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| ''Nine Chapters of Mathematical Arts''1000 BCE? – 50 CE
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| -ch.1, computational algorithm, area of plane figures, GCF, LCD
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| -ch.2, proportions
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| -ch.3, proportions
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| -ch.4, square, cube roots, finding unknowns
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| -ch.5, volume and usage of pi as 3
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| -ch.6, proportions
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| -ch,7, interdeterminate equations
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| -ch.8, Gaussian elimination and matrices
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| -ch.9, Pythagorean theorem (Gougu Theorem)
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| == Mathematics in education ==
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| The first reference to a book being used in learning mathematics in China is dated to the second century CE ([[Hou Hanshu]]: 24, 862; 35,1207). We are told that [[Ma Xu]] (a youth ca 110) and [[Zheng Xuan]] (127-200) both studied the ''Nine Chapters on Mathematical procedures''. C.Cullen claims that mathematics, in a manner akin to medicine, was taught orally. The stylistics of the ''[[Suàn shù shū]]'' from Zhangjiashan suggest that the text was assembled from various sources and then underwent codification.<ref>Christopher Cullen, "Numbers, numeracy and the cosmos" in Loewe-Nylan, ''China's Early Empires'', 2010:337-8.</ref>
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| ==See also==
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| *[[Chinese astronomy]]
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| *[[History of mathematics]]
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| **[[Indian mathematics]]
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| **[[Islamic mathematics]]
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| **[[Japanese mathematics]]
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| == Footnotes and references ==
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| *{{PD-old-text|title=Americanized Encyclopædia Britannica: rev. and amended A dictionary of arts, sciences and literature, to which is added biographies of living subjects. 96 colored maps and numerous illustrations, Volume 9|year=1890|author=}}
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| *{{PD-old-text|title=The home encyclopædia: compiled and revised to date from the leading encyclopædias, Volume 18|year=1895|author=}}
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| *{{PD-old-text|title=Americanized Encyclopædia Britannica, revised and amended: A dictionary of arts, sciences and literature; to which is added biographies of livings subjects ...|year=1890|author=}}
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| *{{PD-old-text|title=The encyclopædia britannica: a dictionary of arts, sciences, literature and general information, Volume 26|year=1911|author=Hugh Chisholm}}
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| *{{PD-old-text|title=The Encyclopædia Britannica: a dictionary of arts, sciences, and general literature, Volume 23|year=1888|author=Thomas Spencer Baynes}}
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| *{{PD-old-text|title=The Encyclopædia Britannica: a dictionary of arts, sciences, literature and general information, Volume 26|year=1911|author=Hugh Chisholm}}
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| *{{PD-old-text|title=The Encyclopædia britannica: a dictionary of arts, sciences, and general literature ; the R.S. Peale reprint, with new maps and original American articles, Volume 23|year=1893|author=William Harrison De Puy}}
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| *{{PD-old-text|title=The Life of the Buddha and the early history of his order: derived from Tibetan works in the Bkah-hgyur and Bstan-hgyur followed by notices on the early history of Tibet and Khoten|year=1907|author=Translated by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio}}
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| *{{PD-old-text|title=The life of the Buddha: and the early history of his order|year=1884|author=William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio}}
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| {{reflist|colwidth=30em}}
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| == Sources ==
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| * {{cite book|last=Boyer |first=C. B. |others= rev. by Uta C. Merzbach|title=A History of Mathematics|edition=2nd |location=New York |publisher=Wiley, |year=1989 |ISBN=0-471-09763-2}} (1991 pbk ed. ISBN 0-471-54397-7)
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| *{{cite book
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| | first=Joseph W.
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| | last=Dauben
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| |authorlink=Joseph Dauben
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| | editor=Victor J. Katz
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| | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
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| | chapter=Chinese Mathematics
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| | publisher=Princeton University Press
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| | year=2007
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| | isbn=978-0-691-11485-9
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| }}
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| * {{cite book|first=Jean-Claude|last=Martzloff|title=A History of Chinese Mathematics|publisher=Springer|isbn=3-540-33782-2|year=1996}}
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| * {{cite book|first=Joseph |last=Needham|title=Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth|location=Taipei|publisher=Caves Books, Ltd.|year=1986}}
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| ==External links==
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| *[http://ctext.org/mathematics Early mathematics texts] (Chinese) - [[Chinese Text Project]]
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| *[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Chinese_overview.html#s31 Overview of Chinese mathematics]
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| *[http://mcel.pacificu.edu/as/students/math/math.htm Chinese Mathematics Through the Han Dynasty]
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| *[http://www.wdl.org/en/item/4721 Primer of Mathematics] by [[Zhu Shijie]]
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| {{S&T in China}}
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| {{DEFAULTSORT:Chinese Mathematics}}
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| [[Category:Chinese mathematics| ]]
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殺意があります
殺意があります！アメージング人！ プラダ人気財布 「この時点で、「ヤンは「最も深刻な警告が来た
3影が、空には雲が低温側で見ているようだ、すべての殺人が濃縮されている prada 財布 通販。最も凶暴な打撃を醸造で釘付けに肉食獣のように プラダ 財布 新作。
第二百十章とげの月、星を破壊し、朱西安 私を殺すために起こっているhuanhangrn「私は大草原を選んだが、私は夜の血を持っていて、身体の王である、それは、私はスクリーンセーバーをマスターすることはできませんが残念ですが、ありません準備せずに、私をスパイするために誰か、ヤンあなたは私のために解くさて！私はわずかなマナに近すぎる、最も重要な瞬間に智リアン超自然的な力を、指し、また瞬間があった、また瞬間があった、私は2番目の頭を開くことができました！ 財布 プラダ レディース '
側冷たい心と魂スターパワーでの護衛の、及び集積闘志遺物、第二の心を開くには、彼の心に渡されたわずかな気晴らしが、ヤンのアイデアを持つことができない、彼はまた感じたが、彼ハート
相关的主题文章：
' この名前、心あなたが怖がって聞いたが、
散乱始めているGangqi。
「よく、彼は最高のMogong木綿で大切除！、大削り取る手術クラス超自然をキャストしたい！カットするようにしてください、そうでないと思ったが、永遠の破壊をカットする必要があります！ プラダ 財布 '
ヤン区日数黒ギザギザ段ボール道路から出てくる水を見悪魔、助けることが驚きませんでした プラダ メンズ ベルト。
'大きな切除 プラダ 財布 値段！' この名前、心あなたが怖がって聞いたが、
サイド寒さが、残余の魂をクリーンアップする悪魔未満のマスター木綿セーバーエクセルので、それは本当に垂直方向と水平方向の世界であることができ、薄い死んだラクダ馬よりも大きいとは思いませんでした。
「Tiens布ギャング！ 財布 プラダ レディース '
ためらい、正方形の冷たい心の動きは、数フィートの厚Gangqi壁への前日の突然3獅子は自分の目の前に抵抗する。
'カット prada トートバッグ！' 牙漢江におけるhuanhangrnはちょうどこのアクションを完了し、水の狂気の悪魔ビッグカット
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to escape out
Energy shot for me, but I do not care, I'll let you prada see that I am プラダ 財布 スタッズ more powerful than the strength of the wind edge, so you know do not disgrace my servant but a glory. '
heart Huang Son look at all this is very プラダ人気財布 accurate, dòng police people.
'Then you try.' bare Senior Gu Changfeng not able to tolerate kneeling here, everything is not able to tolerate it, can not wait to be broken seal, to escape out, プラダ 財布 定価 crazy vent, then penance supernatural prada powers, looking for party cold revenge.
'Well!'
heart Wong Son face sè look turned serious: 'You ready for my magic will directly damage the seal, if you could not stand, it will blown powder.' between
speak, one of his big hands in the air immediately appeared many free runes, these runes formed a huge statue of the heart, like lotus tires in general, that huge heart center
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Grip to the unseen world
Grip to the unseen world, Lent god statue body.
However, a huge fist, as if the dead among プラダ 財布 中古 the powerful spirits Wang broke the eternal seal, shock out of direct bombardment to the cold side of the head! very
dangerous!
this look if the bombardment, the party cold I'm afraid to be beaten to pieces, being eternal repression!
In this critical moment, the cold side of life and death fight many times, have got experience of the reign of terror once again prada perfect show, do not know somewhere, where they come prada 財布 スタッズ from tyrannical body will mobilize all their strength, the strength of all 長財布 prada magic, all together, the moment between a full body close to the cold side of Chunyang Shizhao immortality is burned, not the cost of transporting many elixir out into the prada 財布 2014 Coorong zhenjun other powerful masters of the body.
these experts are taking a 'risk-day battle elixir.' Mana condensed into one. Full body binding and square cold.
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then there are countless lives Great concubine
Find square Kiyoshi senior sister apprentice, and she must sacrifice refining a baby, this baby Ji Lian Well, even if it is prada トートバッグ wishful son brought 'Wild ancient プラダ スタッズ 財布 furnace' confrontation I have no use. '
side cold body flash, flew out from the 'eight Buddha' at home.
Jia blue gently sigh, prada sat down cross-legged and started running saver.
'That Jia Blue senior sister apprentice really interesting to me, but ...... ..........' After 方寒飞 out eight Buddha, secretly sighed.
'What are they afraid! eat her cold side you later do the magic figure of God, three thousand imperial concubines are too few, then there are countless lives Great concubine, プラダ メンズ ベルト and even countless confidante. Even Extreme wind feather palm to teach you , then when there are countless romantic bonds in the body. '
Yan growled figure in their lives.
'eternal giant marble so pradaの財布 I say it, and now I'm going to party Kiyoshi, with her immortal power breaks, refining
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if this can be the eldest of all ages
Han beard suddenly stop down, breathing heavily.
'Well, here we are プラダ最新財布 refining today, anyway, this is almost full power of magic, Ula fellow, it seems your position in points among the firm Baoyan, what material there is nothing material to be even Once upon a time even began some are willing to allocate it, how it hundreds of years among fellow immortal also practicing prada ピンク 財布 to the peak, if this can be the eldest of all ages, space law is certain to be able to practice , it is asking for the moon. '
good woman can stay down, shut the fire, 'I have to go back and rest.'
'I have to go back.' mysterious old man said.
'Well, I プラダ 財布 値段 stayed to see Shoudan furnace, lest other Lian prada メンズ 財布 Po division to peep.' sat down cross-legged on the cold side, プラダ 長財布 silently Breath, 'Now this magic wand, that is temperature dependent, waiting for Missy comes, we the furnace together, came up with the world
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their saver again pulled Abi gas
Net World Lotus! '
the most important thing at this juncture, プラダ新作バッグ2014 'Zhu Xian Pirates' also move, she naturally can not sit and watch 'Star Pirates off' 'Pirates of thorns month,' the saver is missing, you know saver loss, not only of all the supernatural powers are all gone, And the essence of human life, mana will be reduced 99%. That is, the thorn months プラダ 財布 値段 Pirates, Pirates star saver prada トートバッグ off into the door Abi's future, both 財布 プラダ レディース of whom will be severe drop magic, and extreme physical weakness, life greatly reduced.
fingers open, a lotus rotate out, also with the ghost sword prada スタッズ 財布 to kill the snake to the cold side.
baby Xianshibumiao nebula, moon and star bracelet lift up a brace, suddenly propped up a Star cover, the cold side, myself, and even 'off Star Pirates,' 'Pirates of thorns month' all wrapped in one . power of the stars
two thieves are endless a disturbance, suddenly scattered mind again, their saver again pulled Abi gas, gradually to put the gateway to
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he is jealous of pity
There are a lot of treasure, Wang Pin cents. Once looted, the prada リボン 財布 consequences could be disastrous, and there I 財布 プラダ am generally not able to come in! And heaven only ancient imperial many people to discuss, the meeting together, or Heaven's Soldiers behest, before they can enter. '
'hateful ah damn! so quickly put this child in front of eighteen layers looted, valuable things all taken away.'
disaster Wong is also angry rampage three dead prada 財布 新作 god, he is jealous of プラダ最新財布 pity, what a rich treasure of heaven, and it was all looted or プラダ 財布 レディース plundered an enemy, and no one will not be easy.
and disaster Huang usually not able to enter heaven treasure, and that he not get the baby.
Huangpu Avenue This is the first time to enter heaven treasure, envy, heaven is heaven than the treasure trove of Divine Pure Land is much greater, he even wants to plunder something, but no, no guts.
however, saw the cold side, a large fishing
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