14 1. Introduction John Bernoulli (1697b) posed the problem of finding geodesics on convex surfaces. In 1698, he remarked, in a letter to Leibniz, that geodesics always have osculating planes that cut the surface at right angles. (An osculating plane is the plane that passes through three nearby points on a curve as two of these points approach the third point.) This geometric property is frequently used as the definition of a geodesic curve, irrespective of whether the curve actually min- imizes arc length. Later, Euler (1732) derived differential equations for geodesics on surfaces using the calculus of variations. This was Euler’s earliest known use of the calculus of variations. Finding shortest paths is easiest on simple surfaces of revolution. Geodesics on surfaces of revolution satisfy a simple first integral or “conservation law” that was first published by Clairaut (1733). Jacobi (1839), in a tour de force, succeeded in integrating the equations of geodesics for a more complicated surface, a triaxial ellipsoid. 1.4. Minimal surfaces We may minimize areas as well as lengths. Consider two points, y(a) = ya , y(b) = yb , (1.43) in the plane (see Figure 1.8). We wish to join these two points by a continuously differentiable curve, y = y(x) ≥ 0 , (1.44) in such a way that the surface of revolution, generated by rotating this curve about the x-axis, has the smallest possible area S. In other words, we wish to minimize S = 2π b a y(x) 1 + y 2 dx . (1.45) Some of you will recognize this as the “soap-film problem.” Sup- pose we wish to find the shape of a soap film that connects two wire hoops. For a soap film with constant film tension, the surface energy is proportional to the area of the film. Minimizing the surface energy of the film is thus equivalent to minimizing its surface area (Isenberg,

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