We study the best Lipschitz constant of a geographical map known as the Delisle map, a projection of a region of the sphere onto a cone. This map was used by Euler for drawing the map of the Russian empire. Euler considered that this map is the most appropriate, given the extent of the region to draw. We provide numerical evidence to show that the Delisle map is indeed the best among several maps from the sphere to a cone, for the region considered by Euler. The maps we consider are natural maps from the sphere to a cone which are also used in geography. We also discuss the quasiconformal dilatations of these maps.