ho.history overview - Newton and Newton polygon

If memory serves correct the history of Newton's polygon and Puiseaux series has some subtleties, so be a bit wary of secondary historical sources. Histories of mathematics are bursting at the seams with romanticized legends, so it is always best to consult primary sources if you wish to know the real history. The following note from Chrystal's Algebra may serve as a helpful entry into the primary literature.

Historical Note. - As has already been remarked, the fundamental idea of
the reversion of series, and of the
expansion of the roots of algebraical
or other equations in power-series
originated with Newton. His famous"
Parallelogram" is first mentioned in
the second letter to Oldenburg; but is
more fully explained in the
Geometria Analytica (see Horsley's edition of Newton's Works, t. i., p.
398). The method was well understood
by Newton's followers, Stirling and
Taylor; but seems to have been lost
sight of in England after their time.
It was much used (in a modified form
of De Gua's) by Cramer in his
well-known Analyse dea Lignes
Courbea Algebriques (1750). Lagrange
gave a complete analytical form to
Newton's method in his "Memoire sur
l'Usage des Fractions Continues,"
Nouv. Mem. d. l'Ac. roy. d. Sciences d. Berlin (1776). (See OEuvres de
Lagrange, t. iv.)

Notwithstanding its great utility, the
method was everywhere all but
forgotten in the early part of this
century, as has been pointed out by De
Morgan in an interesting account of
it given in the Cambridge
Philosophical Transactions, vol.ix.
(1855).

The idea of demonstrating, a priori,
the possibility of expansions such as
the reversion-formulae of S.18
originated with Cauchy; and to him, in
effect, are due the methods employed
in SS.18 and 19. See his memoirs on
the Integration of Partial
Differential Equations, on the
Calculus of Limits, and on the Nature
and Properties of the Roots of an
Equation which contains a Variable
Parameter,
Exercices d'Analyse et de Physique Mathematique, t. i. (1840), p. 327;
t. ii. (1841), pp. 41, 109. The form
of the demonstrations given in SS. 18,
19 has been borrowed partly from
Thomae, El. Theorie der Analytischen
Functionen einer Complexen
Veranderlichen (Halle, 1880), p. 107;
partly from Stolz,
Allgemeine Arithmetik, I. Th. (Leipzig, 1885), p. 296.

The Parallelogram of Newton was used
for the theoretical purpose of
establishing the expansibility of the
branches of an algebraic function by
Puiseaux in his Classical Memoir on
the Algebraic Functions (Liouv. Math.
Jour., 1850). Puiseaux and Briot and
Bouquet (Theorie des Fonctions
Elliptiques (1875), p. 19) use
Cauchy's Theorem regarding the number
of the roots of an algebraic equation
in a given contour; and thus infer the
continuity of the roots. The
demonstration given in S.21 depends
upon the proof, a priori, of the
possibility of an expansion in a
power-series; and in this respect
follows the original idea of Newton.

The reader who desires to pursue the
subject further may consult Durege,
Elemente der Theorie der Functionen einer Complexen Veranderlichen
Grosse, for a good introduction to
this great branch of modern
function-theory.

The applications are very numerous,
for example, to the finding of
curvatures and curves of closest
contact, and to curve-tracing
generally. A number of beautiful
examples will be found in that
much-to.be-recommended text-book,
Frost's Curve Tracing. -- G. Chrystal: Algebra, Part II, p.370