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发帖数: 4447 | 1 Horner's method
https://en.wikipedia.org/wiki/Horner%27s_method
It is questionable to what extent it was De Morgan's advocacy of Horner's
priority in discovery[5][16] 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)[5][16]
Isaac Newton in 1669 (but precise reference needed)
the Chinese mathematician Zhu Shijie in the 14th century[16]
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[17]
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).[18]
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-quintics, 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.".[19] 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.[20] 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 described
in his book Jigu Suanjing. |
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