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.