# Equation of Catenary/Formulation 2

## Theorem

Consider a **catenary**.

Let a cartesian coordinate plane be arranged so that the y-axis passes through the lowest point of the catenary.

The **catenary** is described by the equation:

- $y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$

where $a$ is a constant.

The lowest point of the chain is at $\tuple {0, a}$.

## Proof

Take the equation of the catenary according to Formulation 1:

- $y = \dfrac {e^{ax} + e^{-ax}} {2 a}$

Put this in a form which uses the hyperbolic cosine:

- $y = \dfrac {\cosh a x} a$

Replace $a$ with $\dfrac 1 a$:

- $y = a \cosh \dfrac x a$

Hence the result.

$\blacksquare$

## Historical Note

The problem of determining the shape of the catenary was posed in $1690$ by Jacob Bernoulli as a challenge.

It had been thought by Galileo to be a parabola.

Huygens showed in $1646$ by physical considerations that it could not be so, but he failed to establish its exact nature.

In $1691$, Leibniz, Huygens and Johann Bernoulli all independently published solutions.

It was Leibniz who gave it the name **catenary**.

From a letter that Johann Bernoulli wrote in $1718$:

*The efforts of my brother were without success. For my part, I was more fortunate, for I found the skill (I say it without boasting; why should I conceal the truth?) to solve it in full ... It is true that it cost me study that robbed me of rest for an entire night. It was a great achievement for those days and for the slight age and experience I then had. The next morning, filled with joy, I ran to my brother, who was still struggling miserably with this Gordian knot without getting anywhere, always thinking like Galileo that the catenary was a parabola. Stop! Stop! I say to him, don't torture yourself any more trying to prove the identity of the catenary with the parabola, since it is entirely false.*

However, Jacob Bernoulli was first to demonstrate that of all possible shapes, the catenary has the lowest center of gravity, and hence the smallest potential energy.

This discovery was significant.

## Lingustic Note

The word **catenary** comes from the Latin word **catena** meaning **chain**.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Catenary: $11.15$