Newton showed that all cubics can be generated by the projection of the five divergent cubic parabolas. Newton's classification of cubic curves appeared in the chapter
"Curves" in Lexicon Technicum by John Harris published in London
in 1710. Newton also classified all cubics into 72 types, missing six of them. In
addition, he showed that any cubic can be obtained by a suitable projection of the
elliptic curve

This is the hardest case and includes the serpentine
curve as one of the subcases. The third class was

(4)

which is called Newton's diverging parabolas. Newton's 66th curve was the trident of Newton.
Newton's classification of cubics was criticized by Euler because it lacked generality.
Plücker later gave a more detailed classification with 219 types.

The nine associated points theorem states that any cubic curve that passes through eight of the nine intersections of
two given cubic curves automatically passes through the ninth (Evelyn et al. 1974,
p. 15).

Pick a point ,
and draw the tangent to the curve at . Call the point where this tangent intersects
the curve .
Draw another tangent and call the point of intersection with the curve . Every curve of third degree has the property that, with the
areas in the above labeled figure,