The study “Chronologia…” published in 1569 testifies to Mercator’s profound knowledge of the science of antiquity (Mercator 1569b). Here Ptolemy is mentioned several times alongside Plato and Hipparchus. In the preface to this work, Mercator noted that he intended to write a work on cosmography. This plan was not realised, but it can be assumed that he did preliminary work on it. It can therefore be assumed that Mercator’s method in creating his projection was conceptually influenced by ancient scholarship and especially by Ptolemy. When attempting to reconstruct Mercator’s methodological approach, the question arose whether the search for a rectilinear representation of the rhumb lines was Mercator’s only intention. The cartographic prerequisites for solving this problem were limited merely to the fact that this required straight lines of meridians arranged in even intervals and likewise straight lines of latitude, but arranged unevenly. It can be assumed that Mercator was also preoccupied with another view of the problem, the solution of which, using only compass and ruler, promised considerably more prospects of success. Mercator was also practically involved with Ptolemaic cartography. For several regional maps, Mercator used the same projection as Ptolemy, in which the meridians appeared as vertical lines and the parallels as horizontal lines. The relation between the meridians and the parallels varied according to latitude. Nevertheless, a good relation could only be achieved with regard to the middle parallel. The length of the middle parallel therefore had to be determined anew from map to map.Footnote 6 This meant that in a regional map with the distance of the parallel from the mid-latitude circle, this relation was distorted both in the direction of the north and in the direction of the south. Consequently, their projection was only useful for mapping relatively small areas. Regional maps with this projection could also not be merged. Thus, a different view of the problem may have emerged for Mercator, which, at least initially, may not have had any direct relation to the straightforward representation of the rhumb lines.
Determining the Length of the Parallel
The problem was to create a map of the world with constant meridian distances, preserving the relations between the meridians and the parallels. This demanded the determination of the length of the parallels. Mercator had two ways of solving this problem, by mathematical calculation or by applying a geometric method. The formulas of spherical geometry required for this and for a simplifying calculation with trigonometric functions were already known to Hipparchus and Ptolemy. Accordingly, Mercator would be able to choose the mathematical calculation. However, he probably chose the more elegant path in Plato’s sense by solving geometric problems with only a compass and ruler. The method for this was already used by Ptolemy in the construction of his projections, but not explicitly described. Recently, this method has been reconstructed (Pápay 2022). Since Mercator was very intensively involved with Ptolemy's theory, it seems very likely that he knew this method, which was based on the transfer the image of the globe onto the flat surface. This enabled him to rectify the circular arc of the parallels. Figures 1 and 2 present this method.
Determining the Increasing Distances of the Parallels
Mercator’s real innovation consisted in working out the geometric method for determining the increasing distances of the parallels. To preserve the relation between the meridian distances and the parallel distances, it was initially necessary to convert the spherical quadrilaterals into rectilinear quadrilaterals. A spherical quadrilateral is a quadrilateral on the globe formed by the circular arcs of two meridians and two parallels. Two circular arcs are meridian segments whose length is constant, in the Fig. 3 it is 5 units corresponding to 5 degrees. There were several possibilities for the transformation of the spherical quadrilaterals into rectilinear quadrilaterals, e.g. they could have been transformed into trapezoids with a perimeter equal to the perimeter of the spherical quadrilaterals. Such a transformation seemed very tempting at first in the attempts to reconstruct Mercator’s method, but then did not lead to the same increasing distances of the parallel as is the case in Mercator’s projection. Further experiments showed that in order to achieve the angular accuracy of his projection, Mercator converted the spherical quadrilaterals into rectilinear quadrilaterals whose perimeter corresponded to the spherical quadrilaterals (Fig. 3). There are several possibilities, but only the following procedure leads to success. The length of the lower parallel and that of the upper parallel are connected and divided into two equal parts. These parts form both the lower and the upper boundary lines of the right-angled quadrilateral. The lateral boundary lines of the quadrilateral are always constant. In Fig. 3, the length is 5°, i.e. 5 units. In this figure, the length of the transformed spherical quadrilateral is 90°, i.e. 90 units. The transformations can be carried out geometrically, i.e. using only a compass and ruler. Mathematical calculations can also be used to perform the transformations. However, Mercator chose the geometric solution with a fairly high degree of certainty, as it corresponded to Plato’s demand for geometry. The written sources mentioned above also suggest a geometrical approach rather than a mathematical calculation.
In the next step, the rectilinear quadrilaterals were enlarged so that their length was unified to the equatorial length, which in Fig. 4 is 90 units corresponding to 90°. The transformation of the quadrilaterals begins at the equator and continues successively in the direction of the pole, as Mercator described: “Having considered the [length of the] latitudes in the direction of the both poles, we have one after the other increased the [length of the] latitudes according to their relation to the equator” (Mercator 1569a). The resulting change in the height of the quadrilaterals determines the increasing distances of the parallels (Fig. 4). The scaled rectification of the spherical quadrilaterals made the rectilinear representation of the rhumb lines possible.
Determining the Parallel Distances Per One Degree
There were several possibilities for Mercator for determining the parallel distances per one degree. It is quite possible that Mercator determined the distances for each degree using the method described above. However, he could also have used a less complicated procedure, the principle of which is shown in Fig. 5. However, one cannot exclude the possibility that Mercator used the rhumb lines for the detailed division of degrees.
Map in the Mercator projection
Figure 6 shows the map in the Mercator projection with the degree grid of Fig. 4. Here it is shown that in this projection, the orthodromes appear as curved lines. The actually spiral-shaped loxodromes, the rhumb lines are shown as straight lines.