A cubic is an algebraic curve of third
Newton distinguished four classes of cubics 1), where each class was divided into several species. He used the theorem that each cubic can be obtained from the divergent parabola, by a central projection 2) from a surface on to another surface.
Newton came to 72 species, including the conic sections 3). This classification was criticized by Euler, because of it's lack of generality. And in fact, Newton missed sex species of curves. There are also other classifications, ranging from 57 to 219 types. The one with the 219 classes was given by Plücker.
Dick Nickalls describes in the Mathematical
Gazette 4 a
new approach to solve the cubic. It gives new geometric insights into how
and why the standard Cardan algebra works.
1) Curves by Sir Isaac Newton in Lexicon Technicum by John Harris, published in London in 1710.
2) Struik 1977 p.138
3) published in 'enumeratio linearum tertii ordinis' (1704)
4) RWD Nickalls (1993): a new approach to solving the cubic Cardan's solution revealed. Mathematical Gazette 1993, volume 77 (November), pp 354 359.