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A type of curve with the equation:
x2y + aby - a2x = 0, where ab > 0
This curve was named and studied by Newton in 1701. It is contained in his classification of cubic curves which appears in "Curves by Sir Isaac Newton in Lexicon Technicum" by John Harris published in 1710. Harris's introduction to the article charmingly states:
"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."
Newton showed 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 are equations of the form
xy2 + ey = ax3 + bx2 + 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 also been studied earlier by L'Hopital and Huygens in 1692.