Curves with Cartesian equation

with . The above equation represents the third class of Newton's classification of Cubic Curves, which Newton divided into five species depending on the Roots of the cubic in on the right-hand side of the equation. Newton described these cases as having the following characteristics:

- 1. ``All the Roots are Real and unequal. Then the Figure is a diverging Parabola of the Form of a Bell, with an Oval at its Vertex.
- 2. Two of the Roots are equal. A Parabola will be formed, either Nodated by touching an Oval, or Punctate, by having the Oval infinitely small.
- 3. The three Roots are equal. This is the Neilian Parabola, commonly called Semi-cubical.
- 4. Only one Real Root. If two of the Roots are impossible, there will be a Pure Parabola of a Bell-like Form''

**References**

MacTutor History of Mathematics Archive. ``Newton's Diverging Parabolas.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Newtons.html.

© 1996-9

1999-05-25