# Definition:Cissoid of Diocles

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## Definition

The **cissoid of Diocles** is the plane curve defined in Cartesian coordinates as:

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

or in polar coordinates as:

- $r = 2 a \sin \theta \tan \theta$

for some real constant $a \in \R_{> 0}$.

The above diagram illustrates the **cissoid of Diocles**.

## Also known as

Some sources refer to the **cissoid of Diocles** as merely a **cissoid**, but that term is best used to refer to the more general object of which the **cissoid of Diocles** was the original instance.

## Source of Name

This entry was named for Diocles of Carystus.

## Historical Note

Diocles of Carystus designed the curve now known as the **cissoid of Diocles** specifically for solving the problem of Doubling the Cube.

It can also be used to divide an angle into any proportion.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**cissoid**

- Weisstein, Eric W. "Cissoid of Diocles." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/CissoidofDiocles.html