The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force. The word catenary is derived from the Latin word for "chain." In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola. The curve is also called the alysoid and chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli. Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690. If you roll a parabola along a straight line, its focus traces out a catenary. As proved by Euler in 1744, the catenary is also the curve which, when rotated, gives the surface of minimum surface area (the catenoid) for the given bounding circle. The integral of the Cartesian equation would be $\displaystyle a^2 \sinh \left ( \frac{x}{a} \right )$ See the gallery below for gallery of places that use a catenary or an inverse catenary. ** Click Read More to View Interactive Graphic ** | Equations: Cartesian Equation: $\displaystyle a\cosh{\frac{x}{a}}$ Parametric Equation: $\displaystyle x=t$ $\displaystyle y=\frac{1}{2}a\left ( e^{\frac{t}{a}}+e^{-\frac{t}{a}} \right )=a\cosh{\frac{t}{a}}$ $\displaystyle \rho a=s^2+a^2$ Graphs: |

**Interactive Graphic:**

**More Info For The Examples:**

**Reference:**

*Catenary*. N.p., Jan. 1999. Web. 06 Dec. 2013.

"Catenary."

*Wikipedia*. Wikimedia Foundation, 12 Feb. 2013. Web. 06 Dec. 2013.

Weisstein, Eric W. "Catenary."

*MathWorld*. Wolfram Research, Inc., 24 Apr. 2003. Web. 6 Dec. 2013.

Weisstein, Eric W. "Cesàro Equation."

*MathWorld*. Wolfram Research, Inc., 26 Oct. 2012. Web. 6 Dec. 2013.