The project that we experimented with was the hanging chain, more properly known as the catenary curve. The motivation for doing this project was simple: we come in contact with catenary curves each and every day, even though we do not realize it. Look at power lines, for example. These are catenary curves. These special curves are defined as "the string whose curve is described by bending under its own weight."1 Well this statement is certainly true for power lines. If you look closely, the line, or string in the definition, does bend due to its own weight.
But how were catenary curves discovered? The answer may surprise you. Catenary comes from the Latin word catena, meaning “chain”2. Even though curves have existed since the beginning of time, it was Galileo who stated first that a curve of a chain hanging under gravity would actually be a parabola, but mathematician Joachim Jungius would later disprove Galileo's theory, stating that it would not be a parabola, but instead another curve (later to be given the title of catenary). But the equation for a catenary curve (the one that we are solving for) actually was obtained by Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli. In fact, it was a challenge that came from Johann's brother, Jacob Bernoulli! But the first person to actually call it a catenary curve was Huygens.3
This project not only made us more aware of differential equations, but more importantly the realization that things in our everyday lives symbolize mathematics in some shape or form.