Horner's Method - History

History

Horner's paper entitled "A new method of solving numerical equations of all orders, by continuous approximation", was read before the Royal Society of London, at its meeting on on July 1, 1819, with Davies Gilbert, Vice-President and Treasurer, in the chair; this was the final meeting of the session before the Society adjorned for its Summer recess. When a sequel was read before the Society in 1823, it was again at the final meeting of the session. On both occasions, papers by James Ivory, FRS, were also read. In 1819, it was Horner's paper that got through to publication in the "Philosophical Transactions". later in the year, Ivory's paper falling by the way, despite Ivory being a Fellow; in 1823, when a total of ten papers were read, fortunes as regards publication, were reversed. But Gilbert, who had strong connections with the West of England and may have had social contact with Horner, resident as Horner was in Bristol and Bath, published his own survey of Horner-type methods earlier in 1823.

Horner's paper in Part II of Philosophical Transactions of the Royal Society of London for 1819 was warmly and expansively welcomed by a reviewer 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. However, the reviewer noted that another, similar method had also recently been promoted by the architect and mathematical expositor, Peter Nicholson. This theme is developed in a further review of some of Nicholson's books in the issue of The Monthly Review for December, 1820, which in turn ends with notice of the appearance of a booklet by Theophilus Holdred, from whom Nicholson acknowledges he obtained the gist of his approach in the first place, although claiming to have improved upon it. The sequence of reviews is concluded in the issue of The Monthly Review for September, 1821, with the reviewer reasserting both Horner's priority and the primacy of his method, judiciously observing that had Holdred published forty years earlier, his contribution could more easily be recognized. The reviewer is exceptionally well-informed, even having sighted Horner's preparatory correspondence with Peter Barlow in 1818, seeking work of Budan. The Bodlean Library, Oxford has the Editor's annotated copy of The Monthly Review from which it is clear that the most active reviewer in mathematics in 1814 and 1815 (the last years for which this information has been published) was none other than Peter Barlow,one of the foremost specialists on approximation theory of the period, suggesting that it was Barlow, who wrote this sequence of reviews. As it also happene, Henry Atkinson, of Newcastle, devised a similar approximation scheme in 1809; he had consulted his fellow Geordie, Charles Hutton, another specialist and a senior colleague of Barlow at the Royal Military Academy, Woolwich, only to be advised that, while his work was publishable, it was unlikely to have much impact. J. R. Young, writing in the mid-1830s, concluded that Holdred's first method replicated Atkinson's while his improved method was only added to Holdred's booklet some months after its first appearance in 1820, when Horner's paper was already in circulation.

The feature of Horner's writing that most distinguishes it from his English contemporaries is the way he draws on the Continental literature, notably the work of Arbogast. The advocacy, as well as the detraction, of Horner's Method has this as an unspoken subtext. Quite how he gained that familiarity has not been determined. Horner is known to have made a close reading of John Bonneycastle's book on algebra. Bonneycastle recognizes that Arbogast has the general, combinatorial expression for the reversion of series, a project going back at least to Newton. But Bonneycastle's main purpose in mentioning Arbogast is not to praise him, but to observe that Arbogast's notation is incompatible with the approach he adopts. The gap in Horner's reading was the work of Paolo Ruffini, except that, as far as awareness of Ruffini goes, citations of Ruffini's work by authors, including medical authors, in Philosophical Transactions speak volumes: there are none - Ruffini's name only appears in 1814, recording a work he donated to the Royal Society. Ruffini might have done better if his work had appeared in French, as had Malfatti's Problem in the reformulation of Joseph Diaz Gergonne, or had he written in French, as had Antonio Cagnoli, a source quoted by Bonneycastle on series reversion (today, Cagnoli is in the Italian Wikipedia, as shown, but has yet to make it into either French or English).

Fuller develops the thesis that Horner's method was never published by Horner until after it was published by Holdred. But this is at variance with the contemporary reception of the works of both Horner and Holdred, as indicated in the previous paragraph, besides the numerous internal flaws in Fuller's paper, flaws that are so strange as to raise doubt as to Fuller's purpose (see the Talk page). Fuller also takes aim at Augustus De Morgan. Precocious though Augustus de Morgan was, he was not the reviewer for The Monthly Review, while several others - Thomas Stephens Davies, J. R. Young - wrote Horner firmly into their records, not least Horner himself, as he published extensively up until the year of his death in 1837. His paper in 1819 was one that would have been difficult to miss. In contrast, the only other mathematical sighting of Holdred is a single named contribution to The Gentleman's Mathematical Companion, an answer to a problem.

It is questionable to what extent it was De Morgan's advocacy of Horner's priority in discovery that led to "Horner's method" being so called in textbooks, but it is true that those suggesting this tend themselves to know of Horner largely through intermediaries, of whom De Morgan made himself a prime example. However, this method qua method was known long before Horner. In reverse chronological order, Horner's method was already known to:

• Paolo Ruffini in 1809 (see Ruffini's rule)
• Isaac Newton in 1669 (but precise reference needed)
• the Chinese mathematician Zhu Shijie in the 14th century
• the Chinese mathematician Qin Jiushao in his Mathematical Treatise in Nine Sections in the 13th century
• the Persian mathematician Sharaf al-Dīn al-Tūsī in the 12th century
• 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 – 220 AD) edited by Liu Hui (fl. 3rd century).

However, this observation on its own masks significant differences in conception and also, as noted with Ruffini's work, issues of accessibility.

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-qintics, of which Qin gives an instance, in keeping with the then Chinese custom of case studies. The first person writing in English to note the connection with Horner's method was Alexander Wylie, writing in The North China Herald in 1852; perhaps conflating and misconstruing different Chinese phrases, Wylie calls the method Harmoniously Alternating Evolution (which does not agree with his Chinese, linglong kaifang, not that at that date he uses pinyin), working the case of one of Qin's quartics and giving, for comparison, the working with Horner's method. Yoshio Mikami in Development of Mathematics in China and Japan published in Leipzig in 1913, gave a detailed description of Qin's method, using the quartic illustrated to the above right in a worked example; he wrote: "who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe ... 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.". However, as Mikami is also aware, it was not altogether impossible that a related work, Si Yuan Yu Jian (Jade Mirror of the Four Unknowns; 1303) by Zhu Shijie might make the shorter journey across to Japan, but seemingly it never did, although another work of Zhu, Suan Xue Qi Meng, had a seminal influence on the development of traditional mathematics in the Edo period, starting in the mid-1600s. Ulrich Libbrecht (at the time teaching in school, but subsequently a professor of comparative philosophy) gave a detailed description in his doctoral thesis of Qin's method, he concluded: It is obvious that this procedure is a Chinese invention....the method was not known in India. He said, Fibonacci probably learned of it from Arabs, who perhaps borrowed from the Chinese. Here, the problems is that there is no more evidence for this speculation than there is of the method being known in India. Of course, 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 he does not specify.