The *fluxion* guy

From: Habitant (berlutte_at_sympatico.ca)
Date: 12/24/03 

Date: Tue, 23 Dec 2003 22:51:33 -0500

maths.ucd.ie/courses/mst3022/history5.pdf

Newton concluded by expressing regret about controversy arising from
his earlier research publications, with the implication that he
intended to publish little in future. Leibniz replied on June 21,
1677. He described his method of drawing tangents, which proceeded
not by fluxions of lines but by differences of numbers. He also
introduced his notation of dx and dy for infinitesimal differences or
differentials of coordinates.

While Leibniz’s differential calculus seems quite familiar to modern
day students of mathematics, the same cannot be said of Newton’s
fluxional calculus, which appears unnecessarily complicated. We will
briefly attempt to describe how Newton’s calculus proceeded. He
assumed that we may conceive of all geometric magnitudes as generated
by continous motion for example, a line is generated by the motion of
a point, etc. The quantity thus generated is called the fluent or
flowing quantity. The velocity of the moving magnitude is the fluxion
of the fluent. There then arise two problems. The first is to find
the fluxion of a given quantity, or more generally the relation of the
fluents being given, to find the relation of their fluxions. This is
akin to implicit differentiation. The second method is the inverse
method of fluxions: from the fluxion, or some relation involving
it, to find the fluent. This amounts to what we call integration,
possibly of a differential equation. Newton referred to these
procedures as the method of quadrature and the inverse
method of tangents. The infinitely small part by which a fluent such
as x increased in a small interval of time o was called the moment of
the fluent and its value was shown to 10 be o . x. Given x, Newton
denoted the fluent whose fluxion was x by x0 or [x]. This is the
same as the integral of x (but not with respect to x). Subsequently,

Newton’s methods were taught at Cambridge for more than 100 years, and
the word fluxion was retained. The explanations given for the
procedures were usually vague and unconvincing. It was
not until almost 1820 that Leibniz’s notation of dx and the integral
sign started to appear in British mathematical textbooks and papers.

Between 1689 and 1693, Newton had fallen under the influence of a
little-known Swiss mathematician named Nicholas Fatio de Duillier
(1664-1753), who had moved to England in 1687. Later in 1693, Newton
suffered what appears to have been a nervous breakdown. After his
recovery, Newton decided to abandon his career in Cambridge,
devoted to research and scientific discovery, and sought more wordly
fame in London as Warden of the Mint in 1696. Nonetheless, Newton was
alarmed when informed by John Wallis in 1695 that many continental
mathematicians considered the calculus to be the invention of Leibniz
alone, perhaps on the basis of his expositions of the subject in Acta
Eruditorum. In support of Newton, Duillier published a paper through
the Royal Society of London in 1699 in which he implied that Leibniz
had plagiarized the ideas of the calculus from Newton. In 1704,
Leibniz replied in the Acta Eruditorum that he had priority of
publication, and he protested to the Royal Society about the
unfairness of the accusation of plagiarism. The Royal Society
eventually responded by establishing a committee to investigate the
dispute. In 1712, the committee published their conclusions in a
report entitled Commercium epistolicum. The report affirmed that

Newton had invented the calculus, a point not seriously disputed even
by Leibniz. The report also noted that Leibniz had had access in the
1670’s to manuscripts and letters describing Newton’s preliminary
version of the calculus (for example, De analysi ), so that suspicions
of plagiarism were not totally dismissed. It has become apparent that
the Commercium epistolicum was essentially Newton’s own work. he
dictated the conclusions that the committee reported.

The priority dispute was dominated by nationalistic concerns, British
mathematicians being especially keen to defend Newton’s honour and
proclaim his genius as a re-flection of British superiority. As a
consequence, the advantages of Leibniz’s notation, subsequently
developed by the Bernoullis and Euler, were ignored in Britain, and by
paying too much deference to Newton, mathematics stagnated there until
the early 19th century. The dispute was led in Britain, albeit
secretly, by Newton himself, and he seems to have wanted to remove all
record of Leibniz’s achievements.
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' As a consequence, the advantages of Leibniz’s notation, subsequently
developed by the Bernoullis and Euler, were ignored in Britain, and by
paying too much deference to Newton, mathematics stagnated there until
the early 19th century.'

LOL