What the Mercator projection does, and why sailors wanted it
In 1569 Gerardus Mercator, a Flemish cartographer working in Duisburg, published a large world map engraved on eighteen sheets. Its title said it was corrected for the use of navigators. The projection it used is now so familiar that it is often mistaken for the shape of the world.
The problem
A ship that holds a constant compass bearing traces a curve on the globe called a rhumb line, or loxodrome, which spirals toward the pole. On the flat charts of the time, such as the portolan charts of the Mediterranean, that course did not come out as a straight line once the voyage was long. A navigator could not lay a ruler between two ports and read off a bearing that would actually get him there.
The construction
Mercator kept every meridian vertical and parallel, which makes the map a rectangle, and widened the spacing between the parallels of latitude as one moves away from the equator. The widening is calculated so that at every point it matches the horizontal stretching that the parallel meridians force on the map. The result is that angles are preserved locally, and every rhumb line becomes a straight line.
The price
The cost is area. Scale grows with the secant of the latitude, so small regions keep their shape while sizes are exaggerated toward the poles. Greenland looks about as large as Africa on a Mercator map, although Africa is roughly fourteen times larger. The poles themselves cannot be shown, because the stretching becomes infinite there.
The mathematics came later
Mercator did not publish how he worked out the spacing of the parallels, and it is thought he used a graphical method. The English mathematician Edward Wright published tables for constructing the projection in 1599, in a book called Certaine Errors in Navigation. The link between the spacing and the integral of the secant was recognised in the seventeenth century.
Sailors took to the projection slowly. It became the standard for sea charts in the eighteenth century, once instruments for measuring latitude and longitude made it practical to plot a ship's position on it.