Editing Horner's method (section)

== History ==
[[File:Qingjiushaoquad1.GIF|thumb|right|200px|[[Qin Jiushao]]'s algorithm for solving the quadratic polynomial equation<math>-x^4+763200x^2-40642560000=0</math><br />result: ''x''=840<ref>{{harvnb|Libbrecht|2005|pages=181–191}}.</ref>]]
Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation",<ref name="Horner">{{harvnb|Horner|1819}}.</ref> was [http://hdl.handle.net/2027/mdp.39015014105277?urlappend=%3Bseq=158 read]  before the Royal Society of London, at its meeting on July 1, 1819, with a sequel in 1823.<ref name="Horner" /> Horner's paper in Part II of ''Philosophical Transactions of the Royal Society of London'' for 1819 was warmly and expansively welcomed by a [http://turing.une.edu.au/~ernie/Horner/Horner1820MonthlyRev91-4.pdf reviewer]{{Dead link|date=January 2020|bot=InternetArchiveBot|fix-attempted=yes}} in the issue of ''The Monthly Review: or, Literary Journal'' for April, 1820; in comparison, a technical paper by [[Charles Babbage]] is dismissed curtly in this review. The sequence of reviews in ''The Monthly Review'' for September, 1821, concludes that Holdred was the first person to discover a direct and general practical solution of numerical equations. Fuller<ref>{{harvnb|Fuller|1999|pages=29–51}}.</ref> showed that the method in Horner's 1819 paper differs from what afterwards became known as "Horner's method" and that in consequence the priority for this method should go to Holdred (1820).

Unlike his English contemporaries, Horner drew on the Continental literature, notably the work of [[Louis François Antoine Arbogast|Arbogast]]. Horner is also known to have made a close reading of John Bonneycastle's book on algebra, though he neglected the work of [[Paolo Ruffini (mathematician)|Paolo Ruffini]].

Although Horner is credited with making the method accessible and practical, it was known long before Horner. In reverse chronological order, Horner's method was already known to:

* [[Paolo Ruffini (mathematician)|Paolo Ruffini]] in 1809 (see [[Ruffini's rule]])<ref name="Cajori">{{harvnb|Cajori|1911}}.</ref><ref name="St Andrews">{{MacTutor|id=Horner}}</ref>
* [[Isaac Newton]] in 1669<ref>Analysis  Per  Quantitatum  Series,  Fluctiones  ac  Differentias : Cum   Enumeratione   Linearum   Tertii   Ordinis,   Londini.      Ex     Officina    Pearsoniana.    Anno  MDCCXI,  p.   10,  4th   paragraph.</ref><ref>Newton's  collected  papers,  the  edition  1779, in  a  footnote,  vol.  I,  p.  270-271</ref>
* the [[Chinese mathematics|Chinese mathematician]] [[Zhu Shijie]] in the 14th century<ref name="St Andrews" />
* the [[Chinese mathematics|Chinese mathematician]] [[Qin Jiushao]] in his ''[[Mathematical Treatise in Nine Sections]]'' in the 13th century
* the [[Persian people|Persian]] [[Mathematics in medieval Islam|mathematician]] [[Sharaf al-Dīn al-Ṭūsī]] in the 12th century (the first to use that method in a general case of cubic equation)<ref>{{harvnb|Berggren|1990|pages=304–309}}.</ref>
* the Chinese mathematician [[Jia Xian]] in the 11th century ([[Song dynasty]])
* ''[[The Nine Chapters on the Mathematical Art]]'', a Chinese work of the [[Han dynasty]] (202 BC&nbsp;– 220 AD) edited by [[Liu Hui]] (fl. 3rd century).<ref>{{harvnb|Temple|1986|p=142}}.</ref>

[[Qin Jiushao]], in his ''Shu Shu Jiu Zhang'' (''[[Mathematical Treatise in Nine Sections]]''; 1247), presents a portfolio of methods of Horner-type for solving polynomial equations, which was based on earlier works of the 11th century Song dynasty mathematician [[Jia Xian]]; for example, one method is specifically suited to bi-quintics, of which Qin gives an instance, in keeping with the then Chinese custom of case studies. [[Yoshio Mikami]] in ''Development of Mathematics in China and Japan'' (Leipzig 1913) wrote:{{Blockquote | style=font-size:100% | text="...&nbsp;who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe&nbsp;... We of course don't intend in any way to ascribe Horner's invention to a Chinese origin, but the lapse of time sufficiently makes it not altogether impossible that the Europeans could have known of the Chinese method in a direct or indirect way."<ref>{{harvnb|Mikami|1913|p=77}}</ref>}}
[[Ulrich Libbrecht]] concluded: ''It is obvious that this procedure is a Chinese invention&nbsp;... the method was not known in India''. He said, Fibonacci probably learned of it from Arabs, who perhaps borrowed from the Chinese.<ref>{{harvnb|Libbrecht|2005|p=208}}.</ref> The extraction of square and cube roots along similar lines is already discussed by [[Liu Hui]] in connection with Problems IV.16 and 22 in ''Jiu Zhang Suan Shu'', while [[Wang Xiaotong]] in the 7th century supposes his readers can solve cubics by an approximation method described in his book [[Jigu Suanjing]].