# Definition:Cissoid of Diocles

## Definition

Let $C$ be a circle of radius $a$ with a distinguished point $O$ on its circumference.

Let $L$ be the tangent to $C$ at the other end of the diameter of $C$ through $O$.

Let $R$ be a point on the circumference of $C$.

Let $OR$ be produced to meet $L$ at $S$.

Let $P$ be the point on $OS$ such that $OP$ = $RS$.

The **cissoid of Diocles** is the locus of points $P$ as $R$ travels around the circumference of $C$.

## 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.

## Also see

- Results about
**the cissoid of Diocles**can be found here.

## 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** in about $180$ BCE, specifically for solving the problem of Doubling the Cube.

It was given its name by Geminus of Rhodes about a century later.

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

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Cissoid of Diocles: $11.34$ - 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): $3$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**cissoid** - 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): $3$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**cissoid** - 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. https://mathworld.wolfram.com/CissoidofDiocles.html