# Definition:Lemniscate of Bernoulli/Cartesian Definition/Also defined as

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## Definition

Some sources define the Cartesian equation for the **lemniscate of Bernoulli** as:

- $\paren {x^2 + y^2}^2 = a^2 \paren {x^2 - y^2}$

which is the same but for a scale factor:

Its associated parametric equation is:

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

## Also see

## Source of Name

This entry was named for Jacob Bernoulli.

## Historical Note

The **lemniscate of Bernoulli** was investigated in some depth by Jacob Bernoulli, from whom it was given its name.

## Linguistic Note

The word **lemniscate** comes from the Latin word **lemniscus**, which means **pendant ribbon**.

The word may ultimately derive from the Latin **lēmniscātus**, which means **decorated with ribbons**.

This may in turn come from the ancient Greek island of **Lemnos** where ribbons were worn as decorations.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Lemniscate: $11.2$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**lemniscate**or**lemniscate of Bernoulli** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**lemniscate** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**lemniscate**