Ernst Zermelo’s first researches in mathematics were in the calculus of variations. His 1894 doctoral dissertation at the University of Berlin extended some of Karl Weierstrass’s methods in the theory of sufficiency. In the years which followed Zermelo’s interests shifted to set theory, and his contributions to this subject would prove to be of fundamental importance.

In 1930 Zermelo returned to the calculus of variations and published two papers on what became known as the navigation problem. His interest in this subject was prompted by the circumnavigation of the globe by the airship "Graf Zeppelin" in 1929. Consider a blimp or plane that moves under power with a given velocity relative to the air, travelling between two points on the earth. Because of the action of wind, the motion of the airship over land is modified. Suppose that the strength and direction of the wind are given as a function of position and time. The problem is to find the trajectory followed by the airship and the corresponding steering angle such that the airship completes its journey in the least time. Following the Hindenburg disaster of 1937, transportation by dirigibles or zeppelins became less common. In later formulations of the problem the airship was often replaced by a boat and the wind by current, and the problem was one of navigation along water.

Zermelo’s solution was based on a special application of the techniques of the calculus of variations, in which he derived a result known as Zermelo’s navigation formula. His result was investigated and extended in the 1930s by such researchers as Tullio Levi-Civita and Constantin Carathéodory. It should be noted that the canonical problems of the calculus of variations - the isoperimetric problem, the hanging chain, the brachistochrone - go back centuries and appear at an early stage in the history of the subject. The navigation problem is somewhat unusual in providing a simple and signature example of very recent vintage, arising from technological developments of the twentieth century.