![Yanghui magic square](https://upload.wikimedia.org/wikipedia/commons/a/ab/Yanghui_magic_square.GIF)
Yang Hui (, ca. 1238–1298),
courtesy name
A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the East Asian cultural sphere, including China, Japan, Korea, and Vietnam.Ulrich Theobald ...
Qianguang (), was a Chinese mathematician and writer during the
Song dynasty
The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the rest ...
. Originally, from Qiantang (modern
Hangzhou
Hangzhou ( or , ; , , Standard Mandarin pronunciation: ), also romanized as Hangchow, is the capital and most populous city of Zhejiang, China. It is located in the northwestern part of the province, sitting at the head of Hangzhou Bay, whi ...
,
Zhejiang
Zhejiang ( or , ; , also romanized as Chekiang) is an eastern, coastal province of the People's Republic of China. Its capital and largest city is Hangzhou, and other notable cities include Ningbo and Wenzhou. Zhejiang is bordered by Jiang ...
), Yang worked on
magic square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
s,
magic circles
A magic circle is a circle of space marked out by practitioners of some branches of ritual magic, which they generally believe will contain energy and form a sacred space, or will provide them a form of magical protection, or both. It may be marke ...
and the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
, and is best known for his contribution of presenting
Yang Hui's Triangle. This triangle was the same as
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
, discovered by Yang's predecessor
Jia Xian
Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty.
Biography
According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...
. Yang was also a contemporary to the other famous mathematician
Qin Jiushao
Qin Jiushao (, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gaug ...
.
Written work
The earliest extant Chinese illustration of '
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
' is from Yang's book ''Xiangjie Jiuzhang Suanfa'' ()
[Fragments of this book was retained in the Yongle Encyclopedia vol 16344, in British Museum Library] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician
Jia Xian
Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty.
Biography
According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...
[Needham, Volume 3, 134-137.] who expounded it around 1100 AD, about 500 years before Pascal. In his book (now lost) known as ''Rújī Shìsuǒ'' () or ''Piling-up Powers and Unlocking Coefficients'', which is known through his contemporary mathematician
Liu Ruxie
/ ( or ) is an East Asian surname. pinyin: in Mandarin Chinese, in Cantonese. It is the family name of the Han dynasty emperors. The character originally meant 'kill', but is now used only as a surname. It is listed 252nd in the classic tex ...
().
[Needham, Volume 3, 137.] Jia described the method used as 'li cheng shi suo' (the tabulation system for unlocking binomial coefficients).
It appeared again in a publication of
Zhu Shijie
Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works ha ...
's book ''Jade Mirror of the Four Unknowns'' () of 1303 AD.
[Needham, Volume 3, 134-135.]
Around 1275 AD, Yang finally had two published mathematical books, which were known as the ''Xugu Zhaiqi Suanfa'' () and the ''Suanfa Tongbian Benmo'' (, summarily called Yang Hui suanfa ).
[Needham, Volume 3, 104.] In the former book, Yang wrote of arrangement of natural numbers around concentric and non concentric circles, known as
magic circles
A magic circle is a circle of space marked out by practitioners of some branches of ritual magic, which they generally believe will contain energy and form a sacred space, or will provide them a form of magical protection, or both. It may be marke ...
and vertical-horizontal
diagram
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...
s of complex
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
arrangements known as
magic squares
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number o ...
, providing rules for their construction.
[Needham, Volume 3, 59-60.] In his writing, he harshly criticized the earlier works of
Li Chunfeng
Li Chunfeng (; 602–670) was a Chinese mathematician, astronomer, historian, and politician who was born in today's Baoji, Shaanxi, during the Sui and Tang dynasties. He was first appointed to the Imperial Astronomy Bureau to help institute a ca ...
and
Liu Yi (), the latter of whom were both content with using methods without working out their theoretical origins or principle.
Displaying a somewhat modern attitude and approach to
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Yang once said:
:''The men of old changed the name of their methods from problem to problem, so that as no specific explanation was given, there is no way of telling their theoretical origin or basis.''
In his written work, Yang provided theoretical proof for the proposition that the complements of the
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
s which are about the diameter of any given parallelogram are equal to one another.
This was the same idea expressed in the Greek mathematician
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's (fl. 300 BC) forty-third proposition of his first book, only Yang used the case of a rectangle and
gnomon
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields.
History
A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the ol ...
.
There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system.
[Needham, Volume 3, 105.] However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian Jesuit
Matteo Ricci
Matteo Ricci, SJ (; la, Mattheus Riccius; 6 October 1552 – 11 May 1610), was an Italians, Italian Society of Jesus, Jesuit Priesthood in the Catholic Church, priest and one of the founding figures of the Jesuit China missions. He create ...
and the
Ming
The Ming dynasty (), officially the Great Ming, was an Dynasties in Chinese history, imperial dynasty of China, ruling from 1368 to 1644 following the collapse of the Mongol Empire, Mongol-led Yuan dynasty. The Ming dynasty was the last ort ...
official
Xu Guangqi
Xu Guangqi or Hsü Kuang-ch'i (April 24, 1562– November 8, 1633), also known by his baptismal name Paul, was a Chinese agronomist, astronomer, mathematician, politician, and writer during the Ming dynasty. Xu was a colleague and collaborato ...
in the early 17th century.
[Needham, Volume 3, 106.]
Yang's writing represents the first in which
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.
[Needham, Volume 3, 46.] Yang was also well known for his ability to manipulate decimal fractions. When he wished to multiply the figures in a rectangular field with a breadth of 24 paces 3
4⁄
10 ft. and length of 36 paces 2
8⁄
10, Yang expressed them in decimal parts of the pace, as 24.68 X 36.56 = 902.3008.
[Needham, Volume 3, 45.]
See also
*
History of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
*
List of mathematicians
Lists of mathematicians cover notable mathematicians by nationality, ethnicity, religion, profession and other characteristics. Alphabetical lists are also available (see table to the right).
Lists by nationality, ethnicity or religion
* List ...
*
Chinese mathematics
Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...
Notes
References
*Needham, Joseph (1986). ''Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth''. Taipei: Caves Books, Ltd.
*Li, Jimin
"Yang Hui" ''
Encyclopedia of China
The ''Encyclopedia of China'' () is the first large-entry modern encyclopedia in the Chinese language. The compilation began in 1978. Published by the Encyclopedia of China Publishing House, the encyclopedia was issued one volume at a time, begin ...
'' (Mathematics Edition), 1st ed.
External links
Yang Hui at MacTutor
{{DEFAULTSORT:Yang, Hui
1238 births
1298 deaths
13th-century Chinese mathematicians
Magic squares
Mathematicians from Zhejiang
Medieval Chinese mathematicians
Song dynasty science writers
Writers from Hangzhou