On the Lambert conformal conical projection and the general map of the Russian Empire

Abstract: The problem of drawing geographical maps is the one of mapping a subset of the sphere, representing a country or some other region on the surface of the Earth, into the Euclidean plane, minimising certain distortion properties that are specified in advance. It is known that from the purely mathematical point of view, this is an extremely difficult problem. One of Leonhard Euler's duties during his first stay at the Imperial Academy of Sciences of Saint Petersburg (1727-1741) was to help establishing maps of the Russian Empire. He worked on this project under the direction of the famous French geographer Joseph-Nicolas Delisle, who was the head of the astronomy and geography departments of the Academy. The general map of the Russian Empire, together with several maps of its particular regions were published under Euler's direction in the so-called Russian Atlas in 1745. In his later memoir ``De proiectione geographica De Lisliana in mappa generali imperii russici usitata'', written in 1777, Euler developed the mathematical theory of the method used by Delisle on a heuristic basis, which he himself used for drawing the general map of the Russian Empire. This method usually carries now the name Delisle--Euler map. In a previous paper, the first two authors of the present paper compared the Delisle--Euler map with several other maps of the conical type, with respect to various mathematical distorsion properties. They showed that this map is the best one from all the points of view considered, when it is applied to the drawing of the Russian Empire. In the present paper, we compare the Euler--Delisle map with a map which was not considered in the paper mentioned, namely, the so-called Lambert conformal conical projection, applied to the same region of the Earth. We show that the latter is better in several respects than all the other maps considered in the previous paper, including the Delisle--Euler map.