In the late 1600’s, Johann Bernoulli came out with a problem to challenge the world’s greatest mathematicians. The problem, known as the Brachistochrone, had garnered some interest and eventually was solved by Johann, Jacob Bernoulli and Newton amongst others. In this talk, we will analyze the solution given by Leonard Euler (and Lagrange), which started a new field of Calculus, called the Calculus of Variations. We will prove the most fundamental equation, Euler-Lagrange, and see how this can be applied to solve the Brachistochrone problem.