**for**

**Background**

A catenary
is the curve formed by a flexible cable of uniform density hanging
from two points under its own weigh. Cables of suspension
bridges and attached to telephone poles hang in this
shape. If the lowest point of the catenary is at
,
then the equation of the catenary
is

.

**Approximated by a parabola**

Notice that is
an even function. The following computation shows that the first term
in the Maclaurin series is . For
this reason it is often claimed that the shape of a hanging cable is
"approximated by a parabola."

.

.

**Arc Length**

The arc length of the
curve is
found by using the integrand . The
length of the catenary over the interval [0,a] is
given by the calculation

.

**Catenary Fit**

In order to find a catenary that has
width and
height all
we need to observe that is
and even function and that
goes through
and we also want it to go through the point . On
first glance, we see that all we need to do is solve the
equation for . However,
this is not possible to do analytically with *Mathematica*, as
we can find out by issuing the following
command.

Therefore, we must resort to using a numerical approximation for c instead of a "formula." In our exploration, the graphs we used the mysterious constant , which was found by determining the catenary that passes through the origin and the point .

**Proof ****The
Catenary** **The
Catenary**

**Computer
Programs** **The
Catenary** **The
Catenary**

**Example 1.** Find the
catenary that goes through the origin and also passes through the two
points with a = 10 and b = 6.

**Solution
1.**

**Example 2.** Find the
equation of the parabola that goes through the origin and the point
. Compare
the parabola solution with the catenary solution.

**Solution
2.**

**Conclusion**

Either a hanging cable is in the shape of a
parabola or a catenary, let's look at the history of this
controversy. The following paragraph is from the "Concise
Encyclopedia of Mathematics" by Eric W. Weisstein.

"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
(MacTutor
Archive). 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." Other mathematicians involved
with the study of the catenary have been Robert
Adrain, James
Stirling, and Leonhard
Euler.

An article suitable for undergraduates to
read is "The
Catenary and the Tractrix (in Classroom Notes)", Robert C.
Yates, American Mathematical Monthly, Vol. 66, No. 6. (Jun. - Jul.,
1959), pp. 500-505.

The St. Louis Arch at the Jefferson National
Expansion Memorial was constructed in the shape of a catenary. Visit
the National Park Service web site arch
history and architecture. Or, go directly
to the web page for the precise mathematical formula for the St.
Louis arch (catenary
curve equation).

The University of British Columbia
Mathematics Department has an amusing
property of the catenary (Java animation). Which is part
of their "Living
Mathematics Project" for "Constructing a new medium for
the communication of Mathematics''.

**Research Experience for
Undergraduates**

**The
Catenary** **The
Catenary** Internet hyperlinks to web sites
and a bibliography of articles.

**Download this
Mathematica Notebook**
**The
Catenary**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004