# Definition:Folium of Descartes

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## Contents

## Definition

### Cartesian Form

The **folium of Descartes** is the locus of the equation:

- $x^3 + y^3 - 3 a x y = 0$

### Parametric Form

The **folium of Descartes** is the locus of the equation given in parametric form as:

- $\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$

## Also see

- Results about
**Folium of Descartes**can be found here.

## Source of Name

This entry was named for René Descartes.

## Historical Note

Marin Mersenne had communicated to René Descartes the method devised by Pierre de Fermat for calculating the tangent to a curve.

René Descartes seems to have thought little of this method, believing that it was not sufficiently general to be useful.

The curve now known as the **folium of Descartes** was used by him as a challenge to Fermat, believing that he would be unable to use this method on it.

Reportedly he was seriously annoyed at Fermat when the latter solved it without any trouble.

## Sources

- Weisstein, Eric W. "Folium of Descartes." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/FoliumofDescartes.html