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{{Short description|British school master and mathematician (1786–1837)}}
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{{other people|William Horner}}
{{distinguish|text=the British biblical scholar [[George William Horner]]}}
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{{Use British English|date=May 2012}}
'''William George Horner''' (9 June 1786 – 22 September 1837) was a [[United Kingdom|British]] [[mathematician]];. Proficient in classics and mathematics, he was a schoolmaster, headmaster and schoolkeeper, proficient in classics as well as mathematics, who wrote extensively on [[functional equationsequation]]s, [[number theory]] and [[approximation theory]], but also on [[optics]]. His contribution to approximation theory is honoured in the designation [[Horner's method]], in particular respect of a paper in ''[[Philosophical Transactions of the Royal Society of London]]'' for 1819. The modern invention of the [[zoetrope]], under the name ''Daedaleum'' in 1834, has been attributed to him.<ref>[httphttps://www.earlycinema.com/zoetrope-motion-picture-technology/zoetrope.html EarlyCinema.comZoetrope]. EarlyCinema.com. Retrieved on 2011-10-11.</ref><ref>[http://wernernekes.de/00_cms/cms/front_content.php?idart=525 Glossary – Z]. Wernernekes.de. Retrieved on 2011-10-11.</ref><ref>{{cite book|title=Philosophical magazine |url=https://books.google.com/books?id=HRebIJSGkJIC&pg=PA36 |accessdate=11 October 2011 |year=1834 |publisher=Taylor & Francis |pagespage=36–36}}</ref>
 
Horner died comparatively young, before the establishment of specialist, regular scientific periodicals. So, the way others have written about him has tended to diverge, sometimes markedly, from his own prolific, if dispersed, record of publications and the contemporary reception of them.
 
==Family life==
The eldest son of the Rev. William Horner, a [[Wesleyan]] minister, Horner was born in [[Bristol]]. He was educated at [[Kingswood School]], a Wesleyan foundation near Bristol, and at the age of sixteen became an assistant master there. In four years he rose to be headmaster (1806), but left in 1809, setting up his own school, The Classical Seminary, at Grosvenor Place, [[Bath, Somerset|Bath]], which he kept until he died there 22 September 1837. He and his wife Sarah (1787?–1864) had six daughters and two sons. One of the sons, another William Horner, continued to run the school. He, too, had a large family; the youngest were twins, Charles and Francis John Horner (1852–1887). Francis Horner matriculated at St. John's College, Cambridge in 1872, taking out a BA in 1876 and an MA in 1883. He became a lecturer in mathematics at the [[University of Sydney|University]] in [[Sydney]], where he died after only a few years - he had been advised to try a change of climate on account of [[tuberculosis]].
 
A longer association with [[Australia]] comes through the issue of Horner's daughter Mary, who retained the name `Horner' through several generations. Mary's son [[Joseph Horner Fletcher]], was a Methodist school headmaster in New Zealand and then Australia. Neville Horner Fletcher (1930–&nbsp;), FTSE, FAA, is a physicist at the Australian National University.
 
On Horner's death in 1837, Sarah Horner lived with another daughter, Charlotte Augusta (1819?–1863; m. 1849), and son-in-law, John {{nowrap|La[u?]mble}} Harrison (1820?–1877), and their daughters, Charlotte Sarah (b. 1852) and Elizabeth Caroline (b. 1856), at 33, Grovesnor Place, Bath
 
Horner's youngest brother, Joseph Horner, was also an assistant master at Kingswood School, but in 1834 matriculated as a mature student at Clare College, Cambridge, standing twelfth Wrangler in the Mathematical Tripos in 1838 (the same year, John Thompson Exley, the son of W. G. Horner's associate [[Thomas Exley]], stood twenty-third). Joseph Horner was a Fellow of Clare College and then vicar of Everton with Tetsworth from 1839 until his death in 1875. He, too, published in mathematics.
 
Other brothers were Thomas Horner, who died young; John Horner, a Wesleyan minister in India; and James Horner, cabinet maker of Bath. According to Horner, John Horner was the first [[missionary]] to come out of Kingswood School: he translated ''Bel and the Dragon'' into [[Marathi language|Marathi]] and his son, Horner's nephew, again John Horner, was tutor to the children of servants in the Sovereign's Household.
 
==Physical sciences, optics==
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as well as to ''Annals of Philosophy'', where Horner begins by responding to other contributors and works up to independent articles of his own; he has a careful style with acknowledgements and, more often than not, cannot resist adding further detail.
 
Several contributions pave the way for, or are otherwise related to, his most celebrated mathematical paper, in ''Philosophical Transactions of the Royal Society of London'' in 1819, which was read by title at the closing meeting for the session on 1 July 1819, with [[Davies Gilbert]] in the Chair. The article, with significant editorial notes by [[Thomas Stephens Davies]], was reprinted as a commemorative tribute in The Ladies' Diary for 1838. The issue of The Gentleman's Diary for that year contains a short obituary notice. A careful analysis of this paper has appeared recently in Craig Smoryński's ''History of Mathematics: A Supplement''.<ref>{{cite book|url=https://books.google.com/books?id=_zliInaOM8UC|title=History of Mathematics: A Supplement|publisher=Springer|place=New York, NY|year=2008|page=|isbn=9780387754802}} esp. Chap. 7</ref>
 
While a sequel was read before the Royal Society, publication was declined for ''Philosophical Transactions'', having to await appearance in a sequence of parts in the first two volumes of ''The Mathematician'' in the mid-1840s, again largely at the instigation of T. S. Davies.
 
However, Horner published on diverse topics in ''The Philosophical Magazine'' well into the 1830s. Davies mooted an edition of Horner's collected papers, but this project never came to fruition, partly on account of Davies' own early death.
 
==Contemporary reception==
Some idea of Horner's standing with his contemporaries is provided by exchanges in the issues of ''Annals of Philosophy'' for July and August, 1817. Thomas Thomson, in commending to an enquirer [[Leonhard Euler|Euler's]] work on algebra, is under some impression that the English translation is by Horner.<ref name=Annals>{{cite journal|title=Annals of Philosophy|editor=Thomas Thomson|date=1817|place=London|volume=Vol. X|hdl = 2027/mdp.39015066710818}}</ref>{{rp|86}} Horner writes promptly to correct this,<ref name=Annals />{{rp|170}} supposing the translation to be the work of [[Peter Barlow (mathematician)|Peter Barlow]]. Thomson, a professor in [[Glasgow]], might not have known that the translation, originally published as far back as 1789, was the work of [[Francis Horner]] MP, an [[Edinburgh]] native, who had died only that February.
 
==Peter Barlow and continued fractions==
When Peter Barlow wrote, in 1845, he remembered Davies, but not Horner, asking to borrow a book by [[François Budan de Boislaurent|Budan]] (both Davies and Horner were living in Bath at the time). Barlow also had a vague recollection that the material on approximations Horner sent him related to [[continued fraction]]s, rather than what appeared in the ''Philosophical Transactions''. Horner clearly held Barlow in high regard and it would have been natural for Horner to approach him to request both books and critical advice as Horner draws attention to Barlow's article in ''New Series of the Mathematical Repository''<ref>{{cite journal|url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433069075566;view=1up;seq=291|title=New Series of the Mathematical Repository|editor=Thomas Leybourn|date=1814|volume=Vol. III}}</ref> and in his survey of approximation methods in the following volume of the Repository (bound up in 1819). The anonymous reviewer for ''The Monthly Review'' in the issue for December, 1820 writes that he has seen Horner's letter to Barlow and that the letter confirms that Horner already had his method of approximation at that date (1818).
 
The methods of both Barlow and Horner use a nesting of expressions akin to continued fractions. Horner was aware of [[Lagrange|Lagrange's]] use of continued fractions at least through his reading of [[John Bonnycastle|Bonnycastle's]] Algebra which is also mentioned in the survey article in the Repository. Horner may have rewritten his paper either under guidance or of his own volition, with an eye to publication in ''Philosophical Transactions''. Horner goes on to write on the use of continued fractions in the summation of series in ''Annals of Philosophy'' in 1826 and on their use in improvements they yield in the solution of equations in ''[[Quarterly Journal of Science, Literature and the Arts]]'' running over into 1827; he explicitly cites work of Lagrange. Barlow's memory of events may have been confused by the appearance of this later work.
 
==Publications==
*''[http://hdl.handle.net/2027/mdp.39015011944330?urlappend=%3Bseq=301 New and important combinations with the Camera Lucida]'', dated Bath, 15 August 1815, Annals of Philosophy, 6 (Oct. 1815), 281-283281–283.
*''[http://hdl.handle.net/2027/mdp.39015010780800?urlappend=%3Bseq=305303 I. On Annuities. - II. Imaginary cube roots. - III. Roots of Binomials]'', dated Bath, 9 September 1816, Annals of Philosophy, 8 (Oct. 1816), 279-284279–284.
*''[http://hdl.handle.net/2027/mdp.39015010780800?urlappend=%3Bseq=418417 Corrections of the paper inserted in the last number of the Annals, p. 279]'', dated Bath, 3 October, 1816Annals1816 Annals of Philosophy, 8 (Nov. 1816), 388-389388–389.
*''[http://hdl.handle.net/2027/mdp.39015066710826?urlappend=%3Bseq=275277 Formulas for estimating the height of mountains]'', dated Bath, 13 February 1817 Annals of Philosophy, 9 (March, 1817), 251--252251–252.
*''[http://hdl.handle.net/2027/mdp.39015066710826?urlappend=%3Bseq=406422 On cubic equations]'', dated Bath, 17 January 1817, Annals of Philosophy, 9 (May, 1817), 378-381378–381.
*''[http://hdl.handle.net/2027/mdp.39015066710818?urlappend=%3Bseq=363 Solution of the equation ${\psi}^{ψ<sup>n}</sup>x=x$]'', Annals of Philosophy, 10 (Nov, 1817), 341-346341–346.
*''[http://hdl.handle.net/2027/mdp.39015013735207?urlappend=%3Bseq=126 On reversion of series, especially in connection with the equation $\psi{\alpha}^{-1}\psi \alpha xψα<sup>−1</sup>ψαx=x$]'', dated Bath, 10 November 1817, Annals of Philosophy, 11 (Feb, 1818), 108-112108–112.
*''[http://hdl.handle.net/2027/nyp.33433069075574?urlappend=%3Bseq=317315 On popular methods of approximation]'', dated Bath, 1819, Math. Rep. New Series, 4 (1819), Part II, 131--136131–136.
* ‘A'A Tribute of Friendship,’ a poem addressed to his friend Thomas Fussell, appended to a ‘Funeral'Funeral Sermon on Mrs. Fussell,’ Bristol, 1820.
*''[http://hdl.handle.net/2027/hvd.hxkkzt?urlappend=%3Bseq=319 On algebraic transformation, as deducible from first principles, and connected with continuous approximations, and the theory of finite and fluxional differences, including some new modes of numerical solution]'', one of ten papers read at the table at the meeting of the Royal on 19 June 1823, immediately before the long vacation adjournment until 20 November 1823; one of the three papers of the set not published in Phil. Trans. that year; published in issues in the first two volumes of The Mathematician bound up in 1845 and 1847.
*''[http://hdl.handle.net/2027/nyp.33433062745116?urlappend=%3Bseq=97 Extension of Theorem of Fermat]'', dated 26 December, Annals of Philosophy New Series, 11 (Feb, 1826), 81-8381–83.
*''On the solutions of the Function ${\psi}^{ψ<sup>z}</sup>x$ and their limitations, [http://hdl.handle.net/2027/nyp.33433062745116?urlappend=%3Bseq=184188 Art 1-8]'', dated Bath, 11 February 1826, Annals of Philosophy New Series, 11 (March, 1826), 168--183168–183; [http://hdl.handle.net/2027/nyp.33433062745116?urlappend=%3Bseq=257261 Art 9-17], ibid, 11 (April, 1826), 241--246241–246.
*''[http://hdl.handle.net/2027/nyp.33433062745116?urlappend=%3Bseq=379 Reply to Mr. Herapath]'', dated Bath, 2 April 1826, Annals of Philosophy New Series, 11 (May, 1826), 363
*''On the use of continued fractions with unrestricted numerators in summation of series, [http://hdl.handle.net/2027/nyp.33433062745116?urlappend=%3Bseq=432436 Art 1-4], dated Bath, 24 April 1826, Annals of Philosophy New Series, 11 (June, 1826), 416–421; [http://hdl.handle.net/2027/nyp.33433062745124?urlappend=%3Bseq=6876 Art 5-6], ibid, 12 (July, 1826), 48-5148–51.
* ‘Natural'Natural Magic,’ a pamphlet on [[optics]] dealing with [[virtual image]]s, London, 1832.
*''On the properties of the Dædaleum, a new instrument of optical illusion'', [http://www.tandfonline.com/doi/abs/10.1080/14786443408648249 Phil. Mag., Ser. 3, 4 (Jan, 1834), 36-41].
[[File:Optics Horner.png|thumb|150px|Frontpage of Horner's 1832 pamphlet on optics]]
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*''On the theory of congeneric surd equations'', Communicated by T. S. Davies, [http://www.tandfonline.com/doi/abs/10.1080/14786443608648795 Phil. Mag., Ser. 3, 8 (Jan, 1836), 43-50].
*''New demonstration of an original proposition in the theory of numbers'', Communicated by T. R. Phillips, [http://www.tandfonline.com/doi/abs/10.1080/14786443708649326 Phil. Mag., Ser. 3, 11 (Nov, 1837), 456-459].
* ‘Questions'Questions for the Examination of Pupils on … General History,’ Bath, 1843, 12mo.
 
A complete edition of Horner's works was promised by [[Thomas Stephens Davies]], but never appeared.
 
==Other contemporary literature==
*P. Barlow, [http://hdl.handle.net/2027/nyp.33433069075566?urlappend=%3Bseq=270 On the resolution of the irreducible case in cubic equations], Math. Rep., NS IV (1814), 46-57 [includes Table for the solution of the irredicibleirreducible case in cubic equations (6pp.)].
*P. Barlow, [http://hdl.handle.net/2027/nyp.33433069075566?urlappend=%3Bseq=291 A new method of approximating towards the roots of equations of all dimensions], Math. Rep., NS IV (1814), No. 12, 67–71.
*T. Holdred, [https://web.archive.org/web/20140106040238/http://turing.une.edu.au/~ernie/Horner/Holdred1820.pdf A New Method of Solving Equations with Ease and Expedition; by which the True Value of the Unknown Quantity is Found Without Previous Reduction. With a Supplement, Containing Two Other Methods of Solving Equations, Derived from the Same Principle](Richard Watts. Sold by Davis and Dickson, mathematical and philosophical booksellers, 17, St. Martin's-le-Grand; and by the author, 2, Denzel Street, Clare-Market, 1820), 56pp..
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*''Register of Kingswood School, 1748-1922'' (1923), p.&nbsp;89.
*1861 Census
 
==External links==
*[http://www.bathbmd.org.uk/index.php Bath: Births, Marriages and Deaths]
*[http://physics.anu.edu.au/people/profile.php?ID=113 Prof. Neville Fletcher, Research School of Physics and Engineering, ANU]
*[http://www.sciencearchive.org.au/scientists/interviews/f/nf.html Australian Academy of Science: Interview with Neville Fletcher]
 
;Attribution
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[[Category:1837 deaths]]
[[Category:Alumni of St John's College, Cambridge]]
[[Category:PeopleScientists from Bristol]]
[[Category:English inventors]]
[[Category:19th-century English mathematicians]]
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