Horner, William George
Horner, William George
(b. Bristol, England, 1786; d. Bath, England, 22 September 1837)
mathematics.
The son of William Horner, a Wesleyan minister, Horner was educated at the Kingswood School, Bristol, where he became an assistant master (stipend £40) at the age of fourteen. After four years he was promoted to headmaster, receiving an additional £10 annually. According to an account given by an “old scholar” in the The History of Kingswood School... By Three Old Boys [A. H. L. Hastings, W. A. Willis, W. P. Workman] (London, 1898), p. 88, the educational regime in his day was somewhat harsh. In 1809 Horner left Bristol to found his own school at Grosvenor Place, Bath, which he kept until his death. He left a widow and several children, one of whom, also named William, carried on the school.
Horner’s only significant contribution to mathematics lay in the method of solving algebraic equations which still bears his name. Contained in a paper submitted to the Royal Society (read by Davies Gilbert on 1 July 1819), “A New Method of Solving Numerical Equations of All Orders by Continuous Approximation,” it was published in the Philosophical Transaction (1819) and was subsequently republished in Ladies’ Diary (1838) and Mathematician (1843). Horner found influential sponsors in J. R. Young of Belfast and Augustus de Morgan, who gave extracts and accounts of the method in their own publications. In consequence of the wide publicity it received, Horner’s method spread rapidly in England but was little used elsewhere in Europe.
Throughout the nineteenth and early twentieth centuries Horner’s method occupied a prominent place in standard English and American textbooks on the theory of equations, although, because of its lack of generality, it has found little favor with modern analysts. With the development of computer methods the subject has declined in importance, but some of Horner’s techniques have been incorporated in courses in numerical analysis.
Briefly, when a real root of an equation has been isolated by any method, it may be calculated by any one of several arithmetical processes. A real root r, of f(x) = 0, is isolated when one finds two real numbers a, b, between which r lies and between which lies no other root of f(x) = 0. Horner’s method consists essentially of successively diminishing the root by the smaller members of successive pairs of positive real numbers.
If
and if x = h + y, we have (expanding by Taylor’s theorem)
If this is written
the coefficients c_{n}, c_{n1}, c_{n2},..., c_{0} in the reduced equation are given by the successive remainders when the given polynomial is divided by (x – h), (x – h)^{2}, (x – h)^{3},..., (x – h)^{n}. In the original account of the method Horner used Arbogast’s derivatives (DϕR, D^{2}ϕR,..., D^{n}ϕR). Later he dispensed altogether with the calculus and gave an account of the method in entirely algebraic terms. Successive transformations were carried out in a compact arithmetic form, and the root obtained by a continuous process was correct to any number of places. The computational schema adopted is often referred to as synthetic division. Horner suggested, correctly, that his method could be applied to the extraction of square and cube roots; but his claims that it extended to irrational and transcendental equations were unfounded.
Although Horner’s method was extremely practical for certain classes of equations, the essentials were by no means new; a similar method was developed by the Chinese in the thirteenth century (see J. Needham, Science and Civilisation in China, I [Cambridge, 1959], p. 42). The iterative method devised by Viète (1600) and developed extensively by Newton (1669), which came to be known as the NewtonRaphson method, is applicable also to logarithmic, trigonometric, and other equations. The numerical solution of equations was a popular subject in the early nineteenth century, and in 1804 a gold medal offered by the Società Italiana delle Scienze for an improved solution was won by Paolo Ruffini (... Sopra la determinazione delle radici... [Modena, 1804]). Ruffini’s method was virtually the same as that developed independently by Horner some years later.
BIBLIOGRAPHY
I. Original Works. Horner’s writings include “A New Method of Solving Numerical Equations of All Orders by Continuous Approximation,” in Philosophical Transactions of the Royal Society, 109 (1819), 308–335; “Horae arithmeticae,” in T. Leybourn, ed., The Mathematical Repository, V, pt. 2 (London, 1830); and “On Algebraic Transformations,” in Mathmatician (1843).
II. Scondary Literature. Accounts of the method are given by J. R. Young in An Elementary Treatise on Algebra (London, 1826); and The Theory and Solution of Algebraical Equations (London, 1843). Augustus de Morgan described the method in sundry articles, including “On Involution and Evolution,” in The Penny Cyclopaedia, vol. XIII (London, 1839); and “Notices of the Progress of the Problem of Evolution,” in The Companion to the Almanack (London, 1839). See also Florian Cajori, “Horner’s Method of Approximation Anticipated by Ruffini,” in Bulletin of the American Mathematical Society, 17 (1911), 409–414.
Margaret E. Baron
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Horner, William George." Complete Dictionary of Scientific Biography. . Encyclopedia.com. 18 Apr. 2019 <https://www.encyclopedia.com>.
"Horner, William George." Complete Dictionary of Scientific Biography. . Encyclopedia.com. (April 18, 2019). https://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/hornerwilliamgeorge
"Horner, William George." Complete Dictionary of Scientific Biography. . Retrieved April 18, 2019 from Encyclopedia.com: https://www.encyclopedia.com/science/dictionariesthesaurusespicturesandpressreleases/hornerwilliamgeorge
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.