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 at the seams full of with 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 helpful entry into the primary literature(and, to boot, Frost's 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. -- G. Chrystal: Algebra, Part II, p.370