In the "Geographike" there are several, partly hidden "mosaic stones" that can be put together to reconstruct Ptolemy's method, when the cartographic approach is given priority. The primary basis for Ptolemy was undoubtedly the comparison with the globe. In his instructions, the globe is mentioned twelve times.Footnote 2 It can be assumed that Ptolemy initially viewed the globe in such a way that the central meridian and the equator appeared as straight lines (Fig. 5). However, he did not transfer the image of the globe onto the flat surface with projecting, but pressed it onto the flat surface virtually.
The great circle of the globe was thereby transformed into a larger circle in which the length of the equator and the middle meridian were each 180 units. In Ptolemy`s instruction, the term "great circle" has two different meanings. First, it refers to the equator, the meridians and in particular the meridians bounding one half of the globe (i.e. the prime meridian together with the meridian of 180°). In a broader sense, it denotes a circle whose radius is 90 units. Such a circle is created by straightening the curved half of the central meridian of 90°, which corresponds to 90 units. In Ptolemy's instruction for his first projection, the transformed great circle does not appear. Why Ptolemy did not describe his procedure and only communicated the results could have several reasons. A very probable reason could be that Ptolemy wanted his instruction to be easy to follow and wanted to avoid any complication.
Based on the structure shown in Fig. 5, it would actually have been possible to create a map network with parallels and meridians. In such a map, however, the oikumene would not be optimally represented. To achieve a better representation of the oikumene, the central parallel of the oikumene had to be moved to the centre. To achieve such a globe image, the globe is to be turn adequately. Ptolemy described such a turning of the globe: “Now when the line of sight is initially directed at the middle of the northern quadrant of the sphere, in which most of the oikimene is mapped, the meridian can give an illusion of straight lines when, by revolving [the globe or the eye] from side to side, each [meridian] stands directly opposite [the eye] and its plane falls through the apex of the sight. The parallels do not do so, however. Because of the oblique position of the north pole [with respect to the viewer]; rather, they clearly give an appearance of circular segments bulging to the south”Footnote 3 (English translation by Berggren and Jones 2000, p. 82). Figure 6 shows such a turning of the globe, which was necessary to create or understand the first projection.
To achieve a greater resemblance to the globe, Ptolemy's basic conception meant that the transformation had to be modified. The rectilinear parallels were to be transformed into circular form. In addition, the representation was to be "tilted" in such a way that the parallel of Rhodes, which in the later phase was also to be made true to the distance, was moved more into the centre of the map. Ptolemy solved these conceptual requirements in such a way that he drew a circle in the transformed great circle for the parallel of Rhodes, which connected the centre of the great circle with the end point of the parallel of Rhodes. Figure 7 presents this method and at the same time provides evidence that Ptolemy determined the distance of the centre of the circles for the parallels from the equator in this way: the cardinal point H is 115 units from the equator. It is very remarkable how ingenious and also elegant was Ptolemy's method in working out his theory of projection.
Very clearly confirms the assumption regarding Ptolemy's methodology the Fig. 8, in which the transformed great circle is combined with the degree network of the first projection.
In the transformed great circle (Fig. 7) only the central meridian and the equator are equidistant. The lengths of the rectilinear parallels are distorted. Figure 9 demonstrates how the exact lengths of the parallels can be determined using the example of Rhodes and Thule.
The diameters of the parallels of Rhodes and Thule on the globe are denoted by a and b. The lengths of the straightened parallels on the half of the globe are designated by a1 and b1. The equations at the bottom right confirm the correctness of the graphical ascertainment of lengths. The following calculations confirm the correctness of the graphical ascertainment of lengths. The length of a multiplied by pi and divided by 2 is equal to the length of a1. The length of b multiplied by pi and divided by 2 is equal to the length of b1.
The determination of the length of the parallel is actually connected with a geometrical problem, with the rectification of the circle, i.e. with the transformation of the circle into a square of equal circumference. Ptolemy's method offered an approximate but very practical solution to this problem. This can be demonstrated with Fig. 9, because half the length of the line a1 corresponds to one side of the square that is circumferentially equal to a circle whose radius is half the length of a. This is a conditional or approximate solution to the otherwise unsolvable geometric problem, but it was perfectly sufficient to create a map projection. It is therefore a conditional or approximate rectification, since the starting point of it is the already rectified great circles, the equator and the central meiridian.
The transformed great circle method allows the ascertainment of correct lengths for each parallel (Fig. 10). The longitude shortens from the equator to the pole in a cosine function. Ptolemy gives the values of the cosine function in his “Almagest”, but without using this term and also without a graphical representation (Stückelberger and Rohner 2012).