Mercator (the Flemish cartographer Gerard de Kremer) produced his famous map projection in 1569. This is a method for mapping the curved surface of the earth on a plane map which is conformal (that is, angles are preserved, and hence shapes are rendered correctly), even though it is not area-preserving. (I remember my school atlas, in which it appeared to a casual glance that Greenland was as big as Africa.)
Historians of science have speculated on how he did it. In summer 2014, I attended a talk by the Portuguese historian Henrique Leitão about the sixteenth-century Portuguese mathematician Pedro Nunes, in which he mentioned research he had done with his colleague Joaquim Alves Gaspar which strongly suggested the method that Mercator used. Now a short article outlining this has appeared in the current European Mathematical Society Newsletter.
In 1537, Nunes investigated rhumb lines or loxodromic curves: these are the lines that a ship would follow if it kept a constant bearing, that is, its course made a constant angle with the meridian. This is obviously very useful for navigation, and his ideas spread rapidly. Several European mathematicians including John Dee worked out tables of rhumb lines. It is relatively straightforward to construct Mercator’s projection from such a table.
Part of the evidence that this is what Mercator actually did is based on careful measurements on his map, and comparisons with tables available to him. These show that the rhumb line solution fits the map more accurately than other methods which have been proposed.
Mercator described his invention as “corresponding to the squaring of a circle in a way that nothing seemed to be lacking save a proof”. Gaspar and Leitão speculate that, by this cryptic remark, he may have meant that he felt sure that his method really was conformal but was unable to prove it. As they say, “the formal demonstration of this property was beyond the reach of mathematics in Mercator’s time”. Perhaps now it would not be beyond a good undergraduate.