Newton's Diverging Parabolas
Curves with Cartesian equation
with
. The above equation represents the third class of Newton's 
classification ofCubic Curves, which Newton divided into five species depending on the Roots of thecubic 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 divergingParabola 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, commonlycalled Semi-cubical.
- 4. Only one Real Root. If two of the Roots are impossible, there will bea 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.
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