What did Newton himself do, so that the "Newton polygon" method is named after him?



Newton and Newton polygon
The Newton polygon and [Newton's method](http://en.wikipedia.org/wiki/Newton%27s_method) are closely related. The following theorem was first proven by Puiseux:



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> if $K$ is an algebraically closed field of characteristic zero, then the field of [Puiseux series](http://en.wikipedia.org/wiki/Puiseux_expansion) over $K$ is the algebraic closure of the field of formal Laurent series over $K$
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However according to Wikipedia



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> This property was implicit in Newton's use of the Newton polygon as early as 1671 and therefore known either as Puiseux's theorem or as the Newton–Puiseux theorem.
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A place where this is illustrated in more detail is "A history of algorithms: from the pebble to the microchip" By Jean-Luc Chabert, Évelyne Barbin, page 191. I will quote the first paragraph



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> Immediately following his description of his numerical method for solving equations, Newton used the same principle to show how to obtain algebraic solutions of equations. He explains how the method of successive linear approximations can be adapted by using a ruler and "small parallelograms", the first version of what is called Newton's polygon. The method was applied in a more general case later, by Puiseux in 1850, both in considering multiple branches and in considering functions of a complex variable
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Then it explains Newton's approach in detail. You can follow the references given there. And then we know the story that this nice tool is now used for the understanding of polynomials over local fields even tough originally the local field was the field of formal Laurent series.


