# Chinese mathematics

Mathematics in China emerged independently by the 11th century BC.[1] The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry.

In the Han Dynasty, the Chinese made substantial progress on root extraction and linear algebra.[2] The major texts from the period,The Nine Chapters on the Mathematical Art and the Writings on Reckoning gave detailed processes to solving mathematical problems in daily life.[3] All procedures were computed using a counting board in both texts, and they included negative numbers as well as fractions. The texts provide procedures similar to that Gaussian elimination and Horner's method for linear algebra and solving quadratic equations, respectively.[4] 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 the method of four unknowns.

As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and 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 for example, 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 Pascal,[5] such as the Song dynasty Chinese polymath Shen Kuo.

## Early Chinese mathematics

Visual proof for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC.
Oracle bone script numeral system
counting rod place value decimal

Simple mathematics on Oracle bone script date back to the Shang Dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the Yi Jing, which greatly influenced written literature during the Zhou Dynasty (1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching (Yi Jing) contained elements of binary numbers.

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] Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry (only Chinese geometry) and the usage of decimals.

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 "Renaissance Man". Six Arts have their roots in the Confucian philosophy.

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.[6] 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."[7] Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved.[7] It stated that two lines of equal length will always finish at the same place,[7] while providing definitions for the comparison of lengths and for parallels,[8] along with principles of space and bounded space.[9] It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.[10] The book provided word recognition for circumference, diameter, and radius, along with the definition of volume.[11]

The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the Zhoubi Suanjing dates around 1200–1000 BC, yet many scholars believed it was written between 300–250 BC. The Zhoubi Suanjing contains an in-depth proof of the Gougu Theorem (a special case of the Pythagorean Theorem) but focuses more on astronomical calculations. However, the recent archaeological discovery of the Tsinghua Bamboo Slips, dated c. 305 BC, has revealed some aspects of pre-Qin mathematics, such as the first known decimal multiplication table.[12]

The abacus was first mentioned in the second century BC, alongside 'calculation with rods' (suan zi) in which small bamboo sticks are placed in successive squares of a checkerboard.[13]

## Qin mathematics

Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars, circa 213–210 BCE. Knowledge of this period can be determined from civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were significant feats of human engineering. Emperor Qin Shihuang （秦始皇） ordered many men to build large, lifesize statues for the palace tomb along with other temples and shrines, and the shape of the tomb was designed with geometric skills of architecture. It is certain that one of the greatest feats of human history, the Great Wall of China, required many mathematical techniques. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion.

Qin bamboo cash purchased at the antiquarian market of Hong Kong by the Yuelu Academy, according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise.

## Han mathematics

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 chousuan, consisting of only nine symbols with a blank space on the counting board representing zero.[2] Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period.[3] The mathematical texts of the time, the Suàn shù shū and the Jiuzhang suanshu solved basic arithmetic problems such as addition, subtraction, multiplication and division.[3] Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order.[4] Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unkowns.[14] The value of pi is taken to be equal to three in both texts.[15] However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used.[3] Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment.[16] The Chinese did not focus on theoretical proofs based on geometry or algebra in the modern sense of proving equations to find area or volume.[17] The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life.[17]

### Suan shu shu

The Suàn shù shū (Writings on Reckoning or The Book of Computations) is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips.[18] It was discovered together with other writings in 1984 when archaeologists 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.[3] 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.[18]

The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art.[18] An example of the 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."[18] Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same excess and deficiency method.[14]

### The Nine Chapters on the Mathematical Art

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.[19] Although the author(s) are unknown, they made a huge contribution in the eastern world. Problems are set up with questions immediately followed by answers and procedure.[16] There are no formal mathematical proofs within the text, just a step by step procedure.[20] The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text.[2]

The Nine Chapters on the Mathematical Art was one of the most influential of all Chinese mathematical books and it is composed of some 246 problems.[19] It would later be incorporated into The Ten Computational Cannons which would become the core of mathematical education in later centuries.[16] This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles.[16] The Nine Chapters made significant additions to solving quadratic equations in a way similar to Horner's Method.[4] It also made advanced contributions to "fangcheng" or what is now known as linear algebra.[14] Chapter seven solves system of linear equations with two unknowns using the excess and deficit method, similar to The Book of Computations.[14] 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.[14] The Nine Chapters solves systems of equations using methods similar to the modern methods of Gaussian elimination and back substitution.[14]

### Calculation of pi

The calculation of pi is often used to gauge the mathematical capabilities of a society. Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area.[19] There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period.[15] Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle.[19] Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154.[3] There is no explicit method or record of how he calculated this estimate.[3]

### Division and root extraction

Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han Dynasty.[3] The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them.[14] Han mathematicians calculated square and cubed roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art.[21] Calculating the squared and cubed roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend (shi) and divisor (fa) throughout the process.[4] This process of successive approximation was then extended to solving quadratics of the second and third order, such as ${\displaystyle x^{2}+a=b}$, using a method similar to Horner's Method.[4] The method was not extended to solve quadratics of the nth order during the Han Dynasty, however, this method would eventually be used to solve these equations.[4]

Fangcheng on a Counting Board

### Linear algebra

The Book of Computations is the first known text to solve systems of equations with two unknowns.[14] There are a total of three sets problems within The Book of Computations involving solving systems of equations with the excess and deficit method, and again are put into practical terms.[14] Chapter Seven of The Nine Chapters on the Mathematical Art also deals with solving a system of two equations with two unknown with the excess and deficit method.[14] To solve for the greater of the two unknowns, the excess and deficit method instructs the reader to cross multiply the minor terms or "zi" (which are the values given for the excess and deficit) with the major terms "mu".[14] To solve for the lesser of the two unknowns, simply add the minor terms together.[14]

Chapter Eight of The Nine Chapters on the Mathematical Art deals with solving infinite equations with infinite unknowns.[14] This process is referred to as the "fangcheng procedure" throughout out the chapter.[14] Many Historians chose to leave the term fangcheng untranslated due to conflicting evidence of what the term means. Many historians refer translate the word to Linear Algebra today. In this chapter, the process of Gaussian Elimination and back substitution are used to solve systems of equations with many unknowns.[14] Problems were done on a counting board and included the use of negative numbers as well as fractions.[14] The counting board was effectively a matrix, where the top line is the first variable of one equation and the bottom was the last.[14]

### Liu Hui's Commentary on The Nine Chapters on the Mathematical Art

Liu Hui's Exhaustion Method

Liu Hui's commentary on The Nine Chapters on the Mathematical Art is the earliest edition of the original text available.[19] Hui is believed by most to be a mathematician shortly after the Han dynasty. Within his commentary, Hui qualified and proved some of the problems from either and algebraic or geometrical standpoint.[17] For instance, throughout The Nine Chapters on the Mathematical Art the value of pi is taken to be equal to three in problems regarding circles or spheres.[15] In his commentary, Liu Hui finds a more accurate estimation of pi using the exhaustion method.[15] The exhaustion method involves creating successive polynomials within a circle so that eventually the area of a higher order polygon will be identical to the circle.[15] From this method, Liu Hui asserted that the value of pi is about 3.14.[3] Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek Method, which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles.[21]

## Mathematics in the period of disunity

Liu Hui's Survey of sea island
Sunzi algorithm for division 400 AD
al Khwarizmi division in the 9th century
Statue of Zu Chongzhi.

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.[22] He was the first Chinese mathematician to calculate π=3.1416 with his 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 and the differential calculus during the 3rd century CE.

fraction interpolation for pi

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 ${\displaystyle {\tfrac {355}{113}}}$ 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"[23]

Along with his son, Zu Geng, Zu Chongzhi used the Cavalieri Method to find an accurate solution for calculating the volume of the sphere. Besides containing formulas for the volume of the sphere, his book also included formulas of cubic equations and the accurate value of pi. 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, 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.

A mathematical manual called "Sunzi mathematical classic" dated between 200—400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. The earliest record of multiplication and division algorithm using the Hindu-Arabic numeral system was in writing by Al Khwarizmi in the early 9th century. Khwarizmi's step by step division algorithm was completely identical to Sunzi division algorithm described in Sunzi mathematical classic four centuries earlier.[24] 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 Sanskrit 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.[25]

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.

## Tang mathematics

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. The Sui dynasty and Tang dynasty ran the "School of Computations".[26]

Wang Xiaotong was a great mathematician in the beginning of the Tang Dynasty, and he wrote a book: Jigu Suanjing (Continuation of Ancient Mathematics), where numerical solutions which general cubic equations appear for the first time[27]

The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of Nam-ri srong btsan, who died in 630.[28][29][30][31][32][33][34][35][36][37]

The table of sines by the Indian mathematician, Aryabhata, were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty.[38] Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas,early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics.[39]

Yi 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.[38] Yi Xing was famed for his genius, and was known to have calculated the number of possible positions on a go board game (though without a symbol for zero he had difficulties expressing the number).

## Song and Yuan mathematics

Northern Song Dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule.[40]

Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD

Four outstanding mathematicians arose during the Song Dynasty and Yuan Dynasty, particularly in the twelfth and thirteenth centuries: Yang Hui, Qin Jiushao, Li Zhi (Li Ye), and Zhu Shijie. Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner-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. 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 Gauss's pivotal condensation.

Qin Jiushao (c. 1202–1261) was the first to introduce the zero symbol into Chinese mathematics.[41] Before this innovation, blank spaces were used instead of zeros in the system of counting rods.[42] 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?"[43] Qin also solved a 10th order equation.[44]

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.[45] Although the Introduction to Computational Studies (算学启蒙) written by Zhu Shijie (fl. 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of Japanese mathematics.[46]

### Algebra

#### Ceyuan haijing

Li Ye's inscribed circle in triangle:Diagram of a round town
Yang Hui's magic concentric circles – numbers on each circle and diameter (ignoring the middle 9) sum to 138

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 (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.[47] "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).

#### Jade Mirror of the Four Unknowns

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 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. It 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.[48]

There are many summation series equations given without proof in the Mirror. A few of the summation series are:[49]

${\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}}$
${\displaystyle 1+8+30+80+\cdots +{n^{2}(n+1)(n+2) \over 3!}={n(n+1)(n+2)(n+3)(4n+1) \over 5!}}$

#### Mathematical Treatise in Nine Sections

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.[47]

#### Magic squares and magic circles

The earliest known magic squares 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.[50] He also worked with magic circle.

### Trigonometry

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.[38] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[38] 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 + 2v2/d, where d is the diameter, v is the versine, c is the length of the chord c subtending the arc.[51] 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).[52] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[38][53] Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, 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).[54]

## Ming mathematics

Ming dynasty selected intellectuals by eight-legged essays in the imperial examinations which neglected and marginalized the development of mathematics. Due to the restriction on new ideologies, especially those confronting Confucianism, the scholars in the Ming dynasty paid less attention to mathematics. Gu Yingxiang and Tang Shunzhi were the leading figures of Chinese mathematics in the Ming dynasty but they knew little about Tian yuan shu or Increase multiply method.[55] In 1450, the Ming mathematician Wu Jing wrote The Annotations of Calculations in the Nine Chapters on the Mathematical Art by collecting and compiling the mathematical calculation problems from the previous dynasties and discarding Tian yuan shu and the increase multiply method.[56]

However, zhusuan, the arithmetic calculation through abacus, gained popularity during the Ming dynasty which was credited with a group of relevant texts. Suanfa Tongzong (General Source of Computational Methods), written by Cheng Dawei (1533-1606) in 1592, stood out and was widely adopted to the development of commerce.[57] This text comprised 17 volumes and 595 entries, presenting a detailed and explicit picture of abacus calculation. Suanfa Tongzong has been reprinted and adapted for over 300 years, enjoying a widespread prestige in the Chinese mathematics society. In the late 16th century, the European missionaries such as Matteo Ricci came to China and translated Euclid's Elements to Chinese with Xu Guangqi.[58] Afterwards, missionaries introduced some elementary mathematical knowledge such as trigonometry and logarithm to China. From then Chinese mathematics started to clash with the west, which contributed to the exchange and collision of mathematical study between China and western countries.

## Qing mathematics

There are many prestigious mathematicians during this period. Mei Wending (1633-1721) devoted himself to the cross-cultural mathematical study and delivered many treaties.[59] His works were compiled by his grandson Mei Goucheng to Meishi Congshu Jiyang (The Compiled works of Mei).[60] Among the 60 volumes of this book, mathematical research takes up 40 volumes, covering nearly all schools of Chinese mathematics at that time and having a profound impact on the further development of Qing mathematics. The Kangxi Emperor had a great interest in mathematics. Under his command, Mei Goucheng and three other outstanding mathematicians compiled a 53-volume Shuli Jingyun (The Essence of Mathematical Study) which gave a systematic introduction of the introduced western mathematical knowledge. This book was officially printed in 1723.[61] The Yongzheng Emperor acceded to the throne in 1723. He believed that the missionary from western countries would endanger his reign and delude his subjects. Therefore, he chased all western missionaries to Macao except some holding office in the royal court. Afterwards, the introduction of western mathematics became stagnant. During this period, Chinese mathematicians were reflecting on the study that they learned from western mathematics on one hand and compiling the classical Chinese mathematical works on another.

## A turning point in Qing mathematics

In 1773, the Qianlong Emperor decided to compile Siku Quanshu (The Complete Library of the Four Treasuries). Dai Zhen (1724-1777) selected and proofread The Nine Chapters on the Mathematical Art from Yongle Encyclopedia and several other mathematical works from Han and Tang dynasties.[62] The long-missing mathematical works from Song and Yuan dynasties such as Si-yüan yü-jian and Ceyuan haijing were also found and printed, which directly led to the wave of investigating and collecting Chinese classical mathematics during the reign of Qianlong Emperor and Jiaqing Emperor.[63] The most annotated work were Jiuzhang suanshu xicaotushuo (The Illustrations of Calculation Process forThe Nine Chapters on the Mathematical Art ) contributed by Li Huang and Siyuan yujian xicao (The Detailed Explanation of Si-yuan yu-jian) by Luo Shilin.[64] They were two extraordinary scholars in Chinese mathematics of that time.

## Late Imperial developments

The word "abacus" was first mentioned by Xu Yue(160–220) in his book suanshu jiyi (算数记遗), or Notes on Traditions of Arithmetic Methods, in Hang Dynasty. As it described, the original abacus had five beads (suan zhu) bunched by a stick in each column, separated by a transverse rod, and arrayed in a wooden rectangle box. One in the upper part represents five and each of four in the lower part represents one. People move the beads to do the calculation.

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.

In 15 century, abacus came into its suan pan form and overtook the counting rods and became the preferred computing device, spreading among merchants to the other countries. It is easy to use and carry, which also provides fast and accurate operation. 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.

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.

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).[65]

A revival of mathematics in China began in the late nineteenth century, when Joseph Edkins, Alexander Wylie and Li Shanlan translated works on astronomy, algebra and differential-integral calculus into Chinese, published by London Missionary Press in Shanghai.

## Post-Imperial Chinese mathematics

In 1840, the First Opium War forced China to open its door and looked at the outside world, which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries. In 1852, the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co-translated the later nine volumes of Elements, 13-volume Algebra and many other western mathematical works.[66] Li Shanlan is the first Chinese mathematician who initiated the modern Chinese mathematical study.[67] However, the post-imperial China witnessed economic intrusion from western countries and domestic social instability, which did not provide a good external environment for Chinese mathematicians to develop their research. The conventional Confucian scholars also had bias against and refused to delve in the western study. Due to the political chaos such as Hundred Days' Reform and New Culture Movement, the research on classical Chinese mathematics was abruptly broken off, and the following Chinese mathematical study was incorporated into the mainstream modern mathematics.

## Modern Chinese mathematics

Chinese mathematics experienced a great surge of revival following the establishment of a modern Chinese republic in 1912. Ever since then, modern Chinese mathematicians have made numerous achievements in various mathematical fields.

Some famous modern ethnic Chinese mathematicians include:

• Shiing-Shen Chern was widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century and was awarded the Wolf prize for his immense number of mathematical contributions.[68][69]
• Ky Fan, made a tremendous number of fundamental contributions to many different fields of mathematics. His work in fixed point theory, in addition to influencing nonlinear functional analysis, has found wide application in mathematical economics and game theory, potential theory, calculus of variations, and differential equations.
• Shing-Tung Yau, his contributions have influenced both physics and mathematics, and he has been active at the interface between geometry and theoretical physics and subsequently awarded the Fields medal for his contributions.
• Terence Tao, an ethnic Chinese child prodigy who received his master's degree at just the age of 16, was the youngest participant in the International Mathematical Olympiad's entire history, first competing at the age of ten, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history. He would go on to receive the Fields medal.
• Yitang Zhang, a number theorist who established the first finite bound on gaps between prime numbers.
• Chen Jingrun, a number theorist who proved that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes) which is now called Chen's theorem .[70] His work was known as a milestone in the research of Goldbach's conjecture.

### Performance at the IMO

In comparison to other participating countries at the International Mathematical Olympiad, China has highest team scores and the won the all-members-gold IMO with a full team the most number of times.[71]

### Abacus-based Mental Calculation

By learning how to calculate with abacus, one can improve his mental calculation which becomes faster and more accurate in doing large number calculations. Abacus‐based mental calculation (AMC) was derived from the abacus which means doing calculation, including addition, subtraction, multiplication, and division, in mind with an imaged abacus. It is a high-level cognitive skill that run through calculations with an effective algorithm. People doing long-term AMC training shows higher numerical memory capacity and has more effectively connected neural pathways.[72][73] They are able to retrieve memory to deal with complex processes to calculate.[74] The processing of AMC involves both the visuospatial and visuomotor processing which generate the visual abacus and perform the movement of the imagery bead.[75] Since the only thing needed to be remembered is the finial position of beads, it takes less memory and less computation time.[75]

## Mathematical texts

Zhou Dynasty

Zhoubi Suanjing c. 1000 BCE-100 CE

• Astronomical theories, and computation techniques
• Proof of the Pythagorean theorem (Shang Gao Theorem)
• Fractional computations
• Pythagorean theorem for astronomical purposes

Nine Chapters on the Mathematical Art 1000 BCE? – 50 CE

• ch.1, computational algorithm, area of plane figures, GCF, LCD
• ch.2, proportions
• ch.3, proportions
• ch.4, square, cube roots, finding unknowns
• ch.5, volume and usage of pi as 3
• ch.6, proportions
• ch,7, interdeterminate equations
• ch.8, Gaussian elimination and matrices
• ch.9, Pythagorean theorem (Gougu Theorem)

Han Dynasty

Book on Numbers and Computation 202 BC-186 BC

• Calculation of the volume of various 3-dimensional shapes
• Calculation of unknown side of rectangle, given area and one side
• Using the false position method for finding roots and the extraction of approximate square roots
• Conversion between different units

## Mathematics in education

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.[76]

## References

### Citations

1. ^ Chinese overview
2. ^ a b c Chemla, Karine. "East Asian Mathematics". Britannica Online Encyclopedia.
3. Needham, Joseph (1959). Science and Civilization in China. England: Cambridge University Press. pp. 1–886. ISBN 0 521 05801 5.
4. Needham, Joseph (1955). "Horner's Method in Chinese Mathematics". T'oung Pao, Second Series. 43: 345–401 – via JSTOR.
5. ^ Frank J. Swetz and T. I. Kao: Was Pythagoras Chinese?
6. ^ Needham, Volume 3, 91.
7. ^ a b c Needham, Volume 3, 92.
8. ^ Needham, Volume 3, 92-93.
9. ^ Needham, Volume 3, 93.
10. ^ Needham, Volume 3, 93-94.
11. ^ Needham, Volume 3, 94.
12. ^ Jane Qiu (7 January 2014). "Ancient times table hidden in Chinese bamboo strips". Nature. Retrieved 15 September 2016.
13. ^ Ifrah, Georges (2001). The Universal History of Computing: From the Abacus to the Quantum Computer. New York, NY: John Wiley & Sons, Inc. ISBN 978-0471396710.
14. Hart, Roger. The Chinese Roots of Linear Alegbra. John's Hopkins University. pp. 11–85. ISBN 978 0801897559.
15. Lennart, Bergren (1997). Pi: A Source Book. New York. ISBN 978-1-4757-2738-8.
16. ^ a b c d Lay Yong, Lam (June 1994). "Nine Chapters on the Mathematical Art: An Overview". Archive for History of Exact Sciences. 47: 1–51 – via Springer.
17. ^ a b c Siu, Man-Keung (1993). "Proof and Pedagogy in Ancient China". Educational Studies in Mathematics. 24: 345–357 – via Springer.
18. ^ a b c d Dauben, Joseph W. (2008). "算数書 Suan Shu Shu A Book on Numbers and Computations: English Translation with Commentary". Archive for History of Exact Sciences. 62 (2): 91–178.
19. Dauben, Joseph (2013). "Nine Chapters on the Art of Mathematics- An Appraisal of the Text, its Editions, and Translations". Sudhoffs Archiv: 199–235 – via JSTOR.
20. ^ Straffin, Philip D. (1998). "Liu Hui and the First Golden Age of Chinese Mathematics". Mathematics Magazine. 71 (3): 163–181. doi:10.2307/2691200.
21. ^ a b Yong, Lam Lay (1970). "The Geometrical Basis of the Ancient Chinese Square-Root Method". Isis. 61 (1): 92–102.
22. ^ 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
23. ^ 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
24. ^ Lam Lay Yong, The Development of Hindu Arabic and Traditional Chinese Arithmetic, Chinese Science, 13 (1996) 35–54
25. ^ Lam Lay Yong, The Development of Hindu Arabic and Traditional Chinese Arithmetic "Archived copy" (PDF). Archived from the original (PDF) on 2012-03-21. Retrieved 2015-12-31.
26. ^ Alexander Karp; Gert Schubring (25 January 2014). Handbook on the History of Mathematics Education. Springer Science & Business Media. pp. 59–. ISBN 978-1-4614-9155-2.
27. ^ Yoshio Mikami, Mathematics in China and Japan,p53
28. ^ 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. Belford-Clarke co. 1890. p. 5826. Retrieved 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
29. ^ The home encyclopædia: compiled and revised to date from the leading encyclopædias, Volume 18. Educational publishing co. 1895. p. 5826. Retrieved 2011-07-01.The Home Encyclopædia: Compiled and Revised to Date from the Leading Encyclopædias
30. ^ Americanized Encyclopædia Britannica, revised and amended: A dictionary of arts, sciences and literature; to which is added biographies of livings subjects ... The "Examiner". 1890. p. 5826. Retrieved 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
31. ^ Hugh Chisholm, ed. (1911). The encyclopædia britannica: a dictionary of arts, sciences, literature and general information, Volume 26 (11 ed.). At the University press. p. 926. Retrieved 2011-07-01.The Encyclopædia Britannica: A Dictionary of Arts, Sciences, Literature and General Information, Hugh Chisholm
32. ^ Thomas Spencer Baynes, ed. (1888). The Encyclopædia Britannica: a dictionary of arts, sciences, and general literature, Volume 23 (9 ed.). C. Scribner's sons. p. 345. Retrieved 2011-07-01.The Encyclopædia Britannica: A Dictionary of Arts, Sciences, and General Literature, Thomas Spencer Baynes
33. ^ Hugh Chisholm (1911). The Encyclopædia Britannica: a dictionary of arts, sciences, literature and general information, Volume 26 (11 ed.). The Encyclopædia Britannica Co. p. 926. Retrieved 2011-07-01.The Encyclopædia Britannica: A Dictionary of Arts, Sciences, Literature and General Information, Hugh Chisholm
34. ^ William Harrison De Puy (1893). 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 (9 ed.). Werner Co. p. 345. Retrieved 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
35. ^ Translated by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio (1907). 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. K. Paul, Trench, Trübner. p. 211. Retrieved 2011-07-01.
36. ^ William Woodville Rockhill; Ernst Leumann; Bunyiu Nanjio (1884). The life of the Buddha: and the early history of his order. Trübner & co. p. 211. Retrieved 2011-07-01.
37. ^ The Life of the Biddha and the Early History of His Order Derived from Tibetan Works in the Bkah-hgyur and Bstan-khoten. Taylor & Francis. p. 211. Retrieved 2011-07-01.
38. Needham, Volume 3, 109.
39. ^ Needham, Volume 3, 108-109.
40. ^ Martzloff, 142
41. ^ Needham, Volume 3, 43.
42. ^ Needham, Volume 3, 62–63.
43. ^ Yoshio Mikami, The development of Mathematics in China and Japan, p77 Leipzig, 1912
44. ^ Ulrich Librecht,Chinese Mathematics in the Thirteenth Century p. 211 Dover 1973
45. ^ Needham, Volume 3, 134–137.
46. ^ Needham, Volume 3, 46.
47. ^ a b (Boyer 1991, "China and India" p. 204)
48. ^ (Boyer 1991, "China and India" p. 203)
49. ^ (Boyer 1991, "China and India" p. 205)
50. ^ (Boyer 1991, "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."
51. ^ Katz, 308.
52. ^ Restivo, 32.
53. ^ Gauchet, 151.
54. ^ Needham, Volume 3, 109–110.
55. ^ He, Ji-Huan (2004-05). "Some interpolation formulas in Chinese ancient mathematics". Applied Mathematics and Computation. 152 (2): 367–371. doi:10.1016/s0096-3003(03)00559-9. ISSN 0096-3003. Check date values in: |date= (help)
56. ^ Jean-Claude., Martzloff, (2006). A history of chinese mathematics. New York: Springer. ISBN 9783540337836. OCLC 262687287.
57. ^ East Asian Journal on Applied Mathematics. Global Science Press.
58. ^ Brucker, Joseph (1912). "Matteo Ricci". The Catholic Encyclopedia. New York: Robert Appleton Company. OCLC 174525342. Retrieved 17 August 2017.
59. ^ 1946-, Elman, Benjamin A., (2005). On their own terms : science in China, 1550-1900. Cambridge, Mass.: Harvard University Press. ISBN 9780674036475. OCLC 443109938.
60. ^ Jami, Catherine (2011-12-01), "A mathematical scholar in Jiangnan: the first half-life of Mei Wending", The Emperor's New MathematicsWestern Learning and Imperial Authority During the Kangxi Reign (1662-1722), Oxford University Press, pp. 82–101, doi:10.1093/acprof:oso/9780199601400.003.0005, ISBN 9780199601400, retrieved 2018-07-28
61. ^ Jami, Catherine; Qi, Han (2003-01-01). "The Reconstruction of Imperial Mathematics in China During the Kangxi Reign (1662-1722)". Early Science and Medicine. 8 (2): 88–110. doi:10.1163/157338203X00026. ISSN 1573-3823.
62. ^ Minghui,, Hu,. China's transition to modernity : the new classical vision of Dai Zhen. Seattle. ISBN 0295741805. OCLC 963736201.
63. ^ Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 ISBN 3-540-33782-2
64. ^ Catherine,, Jami, (2012). The emperor's new mathematics : Western learning and imperial authority during the Kangxi Reign (1662-1722). Oxford: Oxford University Press. ISBN 9780191729218. OCLC 774104121.
65. ^ Needham, Volume 3, 110.
66. ^ Carlyle, Edward Irving (1900). "Wylie, Alexander". In Lee, Sidney. Dictionary of National Biography. 63. London: Smith, Elder & Co.
67. ^ "Li Shanlan's Summation Formulae". A History of Chinese Mathematics: 341–351. doi:10.1007/978-3-540-33783-6_18.
68. ^ "Chern biography". www-history.mcs.st-and.ac.uk. Retrieved 2017-01-16.
69. ^ "12.06.2004 - Renowned mathematician Shiing-Shen Chern, who revitalized the study of geometry, has died at 93 in Tianjin, China". www.berkeley.edu. Retrieved 2017-01-16.
70. ^ J. R., Chen (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.
71. ^
72. ^ Hu, Yuzheng; Geng, Fengji; Tao, Lixia; Hu, Nantu; Du, Fenglei; Fu, Kuang; Chen, Feiyan (2010-12-14). "Enhanced white matter tracts integrity in children with abacus training". Human Brain Mapping. 32 (1): 10–21. doi:10.1002/hbm.20996. ISSN 1065-9471.
73. ^ Wu, Tung-Hsin; Chen, Chia-Lin; Huang, Yung-Hui; Liu, Ren-Shyan; Hsieh, Jen-Chuen; Lee, Jason J. S. (2008-11-05). "Effects of long-term practice and task complexity on brain activities when performing abacus-based mental calculations: a PET study". European Journal of Nuclear Medicine and Molecular Imaging. 36 (3): 436–445. doi:10.1007/s00259-008-0949-0. ISSN 1619-7070.
74. ^ "Brain activation during abacus-based mental calculation with fMRI: a comparison between abacus experts and normal subjects - IEEE Conference Publication". ieeexplore.ieee.org. Retrieved 2018-07-27.
75. ^ a b "Prospective demonstration of brain plasticity after intensive abacus-based mental calculation training: An fMRI study". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 569 (2): 567–571. 2006-12-20. doi:10.1016/j.nima.2006.08.101. ISSN 0168-9002.
76. ^ Christopher Cullen, "Numbers, numeracy and the cosmos" in Loewe-Nylan, China's Early Empires, 2010:337-8.

### Sources

• Boyer, C. B. (1989). A History of Mathematics. rev. by Uta C. Merzbach (2nd ed.). New York: Wiley,. ISBN 0-471-09763-2. (1991 pbk ed. ISBN 0-471-54397-7)
• Dauben, Joseph W. (2007). "Chinese Mathematics". In Victor J. Katz. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 978-0-691-11485-9.
• Lander, Brian. "State Management of River Dikes in Early China: New Sources on the Environmental History of the Central Yangzi Region." T'oung Pao 100.4-5 (2014): 325-62.
• Martzloff, Jean-Claude (1996). A History of Chinese Mathematics. Springer. ISBN 3-540-33782-2.
• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
Public domain
•  This article incorporates text from 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, a publication from 1890 now in the public domain in the United States.
•  This article incorporates text from The home encyclopædia: compiled and revised to date from the leading encyclopædias, Volume 18, a publication from 1895 now in the public domain in the United States.
•  This article incorporates text from Americanized Encyclopædia Britannica, revised and amended: A dictionary of arts, sciences and literature; to which is added biographies of livings subjects ..., a publication from 1890 now in the public domain in the United States.
•  This article incorporates text from The encyclopædia britannica: a dictionary of arts, sciences, literature and general information, Volume 26, by Hugh Chisholm, a publication from 1911 now in the public domain in the United States.
•  This article incorporates text from The Encyclopædia Britannica: a dictionary of arts, sciences, and general literature, Volume 23, by Thomas Spencer Baynes, a publication from 1888 now in the public domain in the United States.
•  This article incorporates text from The Encyclopædia Britannica: a dictionary of arts, sciences, literature and general information, Volume 26, by Hugh Chisholm, a publication from 1911 now in the public domain in the United States.
•  This article incorporates text from 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, by William Harrison De Puy, a publication from 1893 now in the public domain in the United States.
•  This article incorporates text from 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, by Translated by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio, a publication from 1907 now in the public domain in the United States.
•  This article incorporates text from The life of the Buddha: and the early history of his order, by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio, a publication from 1884 now in the public domain in the United States.