In the mid-seventeenth century, it was generally thought that the shape of a hanging chain was a parabola. In a series of letters with Marin Mersenne, the 17-year-old Christiaan Huygens showed that the hanging chain did not in fact take the form of a parabola. We will investigate some of Huygens's geometrical arguments in detail, and we will discuss some of the general history of the problem.