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*