This thesis deals with the subject of mathematical geography. Initially, the two basic problems of mathematical geography are considered in more detail. These fundamental issues are, first, the determination of distance on the earth's surface and secondly to determine the course angle for navigation. The first chapter explores in greater detail the method for drawing solution of the two fundamental problems. The second chapter deals with the geometry on the sphere. In the mathematical terminology, this geometry is called a spherical geometry. In this work, the main terms of the spherical geometry are listed and defined. The spherical distance term and the cosine rule for the spherical geometry will be proved. With its help the two basic problems can be solved mathematically very easily and fast. Chapter 3 presents the various subdivisions of map projections in different groups. A map projection is a projection that maps the points of the earth's surface in the plane. The fourth chapter takes a closer look at the best-known map projection. This is the Mercator projection of Gerardus Mercator, which he published in 1569. It is proved that the Mercator map of the world is equal of angle. The last chapter finally deals with the differential geometry. So first the important properties of IR n be repeated before parameterized curves and surfaces are defined. After some additional definitions the first fundamental form can be defined. Then it is shown that the length of a parameterized curve, the angle between two tangents of the intersecting curves and the size of parametric surfaces depend on the coefficients of the first fundamental form. As a result a criterion for determining the three loyalty properties of general projection can be proved.