Connection between stereometry and planimetry in pre-Euclidean geometry and its application in cartography

The method of the transformed great circle makes it possible to transform the small circles parallel to the great circle of the sphere, i.e. to transform the circles of latitude into straight lines of equal length. This method was not explicitly described by Ptolemy. However, it can be proven that he used this method in the construction of his map projections. As this method does not appear in Euclidean geometry, it has been assumed that it is a much older, pre-Euclidean method. There are no sources for this assumption. One indication of this, however, is that with the method of the transformed great circle, the division of the circle into 360 degrees is only possible geometrically, i.e. only with linear and compass. A further indication of this is the assumption that the planimetric unit circle emerged from the transformed great circle. Certain trigonometric functions in the transformed great circle have a concrete relationship to the sphere, which has been lost in Euclidean geometry through abstraction. The approximate geometric determination of π was also still possible with this original method. In antiquity, the transformed great circle was only used in Ptolemy's cartographic theory of projection. Mercator reduced the transformed great circle to a planimetric unit circle while retaining its geometric properties. This meant that he was able to create his map projection of 1569, which was later calculated using differential calculus, exclusively geometrically.