# Definition:Folium of Descartes/Cartesian Form

## Definition

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

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

## Also presented as

The equation defining the **folium of Descartes** can also be presented as:

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

## Also see

- Results about
**the 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 in $1638$ 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.

## Linguistic Note

The word **folium** in the term **folium of Descartes** derives from the Latin for **leaf**, from the leaf-shaped loop that it encloses.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Folium of Descartes: $11.24$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**folium of Descartes** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**folium of Descartes** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates: Cartesian coordinates

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