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 their seams full of romanticized legends so it is always best to consult primary sources if you wish to know the real history. The following historical note from  Chrystal's *Algebra, Part II, p.370* may serve as a useful entry into the primary literature (and, to boot, places to find beautiful examples of curve tracing - how can you resist?)

**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*.