Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves into 78 species (he published 72; 6 more were found after him): http://www.math.yorku.ca/~steprans/Courses/2041-42/CubicCurves/CubicCurves.shtml

My question: Has there been any sort of generalization to this, for quartics, quintics and beyond? Also, is there anything special about the sequence of numbers that arise? : 3 for quadratics, 78 for cubics, etc., i.e. 3,78,... I would suspect that such a sequence has ties with other branches of maths, albeit underlying, deep and unexpected. Maybe undiscovered.

Thanks!

• A google search for "Newton's enumeration" {AND} "quartic curves" seems to bring up some relevant items. See also the mathoverflow question Newton and Newton polygon for possible suggestions on other things to look at or use in google searches. – Dave L. Renfro Oct 3 '14 at 15:42
• Never mind. $($I was under the mistaken impression that you were the one who brought up this topic in a previous question, which was asked several days ago$)$. Sorry for the confusion. – Lucian Oct 3 '14 at 17:37
• @AnalysisIncarnate "conics, ellipses and hyperbolas" Ellipse and hyperbola are both conic sections. Perhaps you mean "parabolas, ellipses and hyperbolas"? – Balarka Sen Oct 4 '14 at 8:54