**Parametric Cartesian equation: **

*x*^{2}*y* + *aby* - *a*^{2}*x* = 0, *ab* > 0

**Click below to see one of the Associated curves.**

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This curve was named and studied by Newton in 1701. It is contained in his classification of cubic curves which appears in

*The incomparable Sir Isaac Newton gives this following Ennumeration of Geometrical Lines of the Third or Cubick Order; in which you have an admirable account of many Species of Curves which exceed the Conick-Sections, for they go no higher than the Quadratick or Second Order.*

[Here the older spelling has been preserved including *conick*, *Quadratick* and *ennumeration*].

Newton shows that the curve *f*(*x*, *y*) = 0, where *f*(*x*, *y*) is a cubic, can be divided into one of four normal forms. The first of these is equations of the form

xy^{2}+ey=ax^{3}+bx^{2}+cx+d.

This is the hardest case in the classification and the serpentine is one of the subcases of this first normal form.

The serpentine had been studied earlier by de L'Hôpital and Huygens in 1692.

JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Serpentine.html