Serpentine

Cartesian equation: x^2y+aby-a^2x = 0, ab > 0

Click below to see one of the Associated curves.

Definitions of the Associated curves

Evolute       Involute 1       Involute 2

 Inverse curve wrt origin      Inverse wrt another circle

Pedal curve wrt origin      Pedal wrt another point

Negative pedal curve wrt origin      Negative pedal wrt another point

   Caustic wrt horizontal rays      Caustic curve wrt another point
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 London in 1710. Harris' introduction to the article 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.
[Here the older spelling has been preserved including conick 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.

Welcome page                         Famous curves index

Previous curve       (Famous curves)          Next curve

Biographical Index                          Chronologies

History Topics Index                      Birthplace Map

Mathematicians of the day     Anniversaries for the year

Search Form     Simple Search Form    Search Suggestions
JOC/EFR February 1996