Geometry of the Mercator projection from 1569

The discovery that Mercator used a special geometric method in the construction of his projection of 1569 opened up the possibility of an exact geometric construction of such a projection, in which the rhumb lines appear rectilinear. The application of this special geometric method, which could be called "planimetric spherics", does not require the use of either π or trigonometric functions. Its foundations were already implicit in Ptolemaic projection theory. Mercator rediscovered them and developed them further. However, he did not describe his geometric method. As a result, this ingenious and simple method fell into oblivion. This article contains an attempt to reconstruct his method.