TITLE: Newton and Newton polygon
QUESTION [29 upvotes]: What did Newton himself do, so that the "Newton polygon" method is named after him?

REPLY [6 votes]: This was intended to be a comment on Bill Dubuque's answer, but I apparently don't yet have enough reputation points to comment, and in any event this is probably too long to appear as a comment.
Given Chrystal's intended audience, I'm surprised that he didn't mention Talbot's 1860 English translation and extensive commentary of Newton's Enumeration Linearum Tertii Ordinis. In Talbot's work, which is freely available on the internet, see the sections On the Analytical Parallelogram (pp. 88-104) and Examples (pp. 104-112). By the way, whoever scanned the book for google wasn't paying attention when the lengthy list of figures at the end of the book were being scanned, so I'm also giving the University of Michigan Historical Math Collection version, which has those figures correctly scanned.
Sir Isaac Newton's Enumeration of Lines of the Third Order, Generation of Curves by Shadows, Organic Description of Curves, and Construction of Equations by Curves, Translated from the Latin, with notes and examples, by C.R.M. Talbot, 1860.
http://books.google.com/books?id=6I97byFB3v0C
http://name.umdl.umich.edu/ABQ9451.0001.001