Editing Pascal's triangle (section)

== History ==
[[File:Yanghui triangle.gif|thumb|right|upright=1|[[Yang Hui]]'s triangle, as depicted by the Chinese using [[Counting rods|rod numerals]], appears in [[Jade Mirror of the Four Unknowns]], a mathematical work by [[Zhu Shijie]], dated 1303.]]
[[File:TrianguloPascal.jpg|thumb|right|upright=1.25|[[Blaise Pascal|Pascal]]'s version of the triangle]]

The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. The Persian mathematician [[Al-Karaji]] (953–1029) wrote a now-lost book which contained the first description of Pascal's triangle.<ref>{{Cite book|url=https://books.google.com/books?id=kt9DIY1g9HYC&q=al+karaji+pascal%27s+triangle&pg=PA132|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|last=Selin|first=Helaine|date=2008-03-12|publisher=Springer Science & Business Media|isbn=9781402045592|language=en|page=132|bibcode=2008ehst.book.....S|quote=Other, lost works of al-Karaji's are known to have dealt with inderterminate algebra, arithmetic, inheritance algebra, and the construction of buildings. Another contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Sama'wal's Bahir (twelfth century) which heavily drew from the Badi.}}</ref><ref>{{Cite book |last=Rashed |first=R. |url=https://books.google.com/books?id=vSkClSvU_9AC&pg=PA62 |title=The Development of Arabic Mathematics: Between Arithmetic and Algebra |date=1994-06-30 |publisher=Springer Science & Business Media |isbn=978-0-7923-2565-9 |pages=63 |language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=kAjABAAAQBAJ&q=al+karaji+binomial+theorem&pg=PA54|title=From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren|last1=Sidoli|first1=Nathan|last2=Brummelen|first2=Glen Van|date=2013-10-30|publisher=Springer Science & Business Media|isbn=9783642367366|language=en|page=54|quote=However, the use of binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly the table was a local discovery - most probably of al-Karaji."}}</ref> In India, the ''[[Chandaḥśāstra]]'' by the Indian poet and mathematician [[Piṅgala]] (3rd or 2nd century BC) somewhat cryptically describes a method of arranging two types of syllables to form [[metre (poetry)|metre]]s of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator [[Halāyudha]] his "method of pyramidal expansion" (''meru-prastāra'') for counting metres is equivalent to Pascal's triangle.<ref>{{cite journal |last=Alsdorf |first=Ludwig |year=1991 |orig-year=1933 |title=The Pratyayas: Indian Contribution to Combinatorics |journal=Indian Journal of History of Science |volume=26 |number=1 |pages=17–61 |url=https://insa.nic.in/(S(f0a4mvblfb5vfmialsdau2an))/writereaddata/UpLoadedFiles/IJHS/Vol26_1_3_SRSarma.pdf}} Translated by S. R. Sarma from {{cite journal |last=Alsdorf |first=Ludwig |display-authors=0 |title={{mvar|π}} Die Pratyayas. Ein Beitrag zur indischen Mathematik |journal=Zeitschrift für Indologie und Iranistik |volume=9 |year=1933 |pages=97–157 }} {{pb}} {{cite journal |last=Bag |first=Amulya Kumar |title=Binomial theorem in ancient India |journal=Indian Journal of History of Science |volume=1 |number=1 |year=1966 |pages=68–74 |url=http://repository.ias.ac.in/70374/1/10-pub.pdf }} {{pb}} Tertiary sources: {{pb}} {{cite book |title=A Concise History Of Science In India |year=1971 |publisher=Indian National Science Academy |editor-last=Bose |editor-first=D. M. |editor-link=Debendra Mohan Bose |last=Sen |first=Samarendra Nath |chapter=Mathematics |chapter-url=https://archive.org/details/in.ernet.dli.2015.502083/page/n178 |at=Ch. 3, {{pgs|136–212}}, esp. "Permutations, Combinations and Pascal Triangle", {{pgs|156–157}} }} {{pb}} {{cite journal |title=The Binomial Coefficient Function |last=Fowler |first=David H. |author-link= David Fowler (mathematician) |journal=The American Mathematical Monthly |year=1996 |volume=103 |number=1 |pages=1–17, esp. §4 "A Historical Note", {{pgs|10–17}} |doi=10.2307/2975209 |jstor=2975209 }}</ref> It was later repeated by [[Omar Khayyám]] (1048–1131), another Persian mathematician; thus the triangle is also referred to as '''Khayyam's triangle''' ({{langx|fa|مثلث خیام|label=none}}) in Iran.<ref>{{cite book |author=Kennedy, E. |title=Omar Khayyam. The Mathematics Teacher 1958 |jstor=i27957284|year=1966 |publisher=National Council of Teachers of Mathematics |pages=140–142}}</ref> Several theorems related to the triangle were known, including the [[binomial theorem]]. Khayyam used a method of finding [[nth root|''n''th roots]] based on the binomial expansion, and therefore on the binomial coefficients.<ref name=":0">{{citation
 | last = Coolidge | first = J. L. | author-link = Julian Coolidge
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2305028
 | mr = 0028222
 | pages = 147–157
 | title = The story of the binomial theorem
 | volume = 56
 | issue = 3 | year = 1949| doi = 10.2307/2305028 }}.</ref>

Pascal's triangle was known in China during the 11th century through the work of the Chinese mathematician [[Jia Xian]] (1010–1070). During the 13th century, [[Yang Hui]] (1238–1298) defined the triangle, and it is known as '''Yang Hui's triangle''' ({{lang-zh|s=杨辉三角|t=楊輝三角|labels=no}}) in China.<ref>Weisstein, Eric W. (2003). ''CRC concise encyclopedia of mathematics'', p. 2169. {{isbn|978-1-58488-347-0}}.</ref>

In Europe, Pascal's triangle appeared for the first time in the ''Arithmetic'' of [[Jordanus de Nemore]] (13th century).<ref>{{cite journal |last1=Hughes |first1=Barnabas |title=The arithmetical triangle of Jordanus de Nemore |journal=Historia Mathematica |date=1 August 1989 |volume=16 |issue=3 |pages=213–223 |doi=10.1016/0315-0860(89)90018-9 |doi-access=free }}</ref>
The binomial coefficients were calculated by [[Gersonides]] during the early 14th century, using the multiplicative formula for them.<ref name="ed-cam">{{citation|contribution=The arithmetical triangle|first=A. W. F.|last=Edwards|pages=166–180|title=Combinatorics: Ancient and Modern|publisher=Oxford University Press|year=2013|editor1-first=Robin|editor1-last=Wilson|editor2-first=John J.|editor2-last=Watkins}}.</ref> [[Petrus Apianus]] (1495–1552) published the full triangle on the [[Book frontispiece|frontispiece]] of his book on business calculations in 1527.<ref>{{citation|title=Nature of Mathematics|first=Karl J.|last=Smith|publisher=Cengage Learning|year=2010|isbn=9780538737586|page=10|url=https://books.google.com/books?id=Di0HyCgDYq8C&pg=PA10}}.</ref> [[Michael Stifel]] published a portion of the triangle (from the second to the middle column in each row) in 1544, describing it as a table of [[figurate number]]s.<ref name="ed-cam"/> In Italy, Pascal's triangle is referred to as '''Tartaglia's triangle''', named for the Italian algebraist [[Nicolo Tartaglia|Tartaglia]] (1500–1577), who published six rows of the triangle in 1556.<ref name="ed-cam"/> [[Gerolamo Cardano]] also published the triangle as well as the additive and multiplicative rules for constructing it in 1570.<ref name="ed-cam"/>

Pascal's {{lang|fr|Traité du triangle arithmétique}} (''Treatise on Arithmetical Triangle'') was published posthumously in 1665.<ref>{{Cite book |last=Pascal |first=Blaise (1623-1662) Auteur du texte |url=https://gallica.bnf.fr/ark:/12148/btv1b86262012/f1.image |title=Traité du triangle arithmétique , avec quelques autres petits traitez sur la mesme matière. Par Monsieur Pascal |date=1665 |language=EN}}</ref>   In this, Pascal collected several results then known about the triangle, and employed them to solve problems in [[probability theory]]. The triangle was later named for Pascal by [[Pierre Raymond de Montmort]] (1708) who called it {{lang|fr|table de M. Pascal pour les combinaisons}} (French: Mr. Pascal's table for combinations) and [[Abraham de Moivre]] (1730) who called it {{lang|la|Triangulum Arithmeticum PASCALIANUM}} (Latin: Pascal's Arithmetic Triangle), which became the basis of the modern Western name.<ref>{{Cite journal | doi = 10.2307/2975209 | title = The Binomial Coefficient Function | first =  David | last = Fowler | author-link = David Fowler (mathematician) | journal = [[The American Mathematical Monthly]] | volume = 103 | issue = 1 |date=January 1996 | pages = 1–17 | jstor = 2975209 }} See in particular p. 11.</ref>