Gerhard Kremer, born in Flanders in 1512, was interested in everything: philosophy, theology, geography, mathematics, astronomy, printmaking, calligraphy, etc. In 1544 he was imprisoned for heresy. In 1569 he published the world map which made his name famous
How? What? You don’t know the Kremer’s map? That’s because he Latinized his name, which means “merchant”, in Mercator. Do you know the Mercator map? Well done! This card is a true symbol of the era of great discoveries.
Each point on Earth can be identified by its latitude, which varies from 90 ° south to 90 ° north, and its longitude, which varies between 180 ° west and 180 ° east. A very naive idea to map the Earth is to represent a point on the Earth by the point on the map having the same coordinates. On the abscissa the longitude and on the ordinate the latitude. If you combine a degree with a millimeter, for example, you get a rectangular world map that is 36 cm wide and 18 cm high. Meridians and parallels are represented by straight, vertical and horizontal segments. This card has almost no interest except the fact that it is the first one to think of … It is called the flat square.
What is the problem with the flat square? Observe that all the meridians have the same length on Earth. The square flat behaves well from this point of view: the meridians are indeed all represented by vertical segments of the same length. Now observe the parallels on Earth. Their lengths decrease as we go from the equator to the small parallels around the poles. However, on the square flat, all the parallels measure 36 cm, regardless of the latitude
In other words, the square plate respects the lengths along the meridians but not those of the parallels. A very small disk on Earth is distorted on the map into a small ellipse, wider than it is tall.
Mercator wants to solve this problem. He probably understands that he cannot keep both the lengths and the shape of the small disks on all the elements of the map. He then gives himself the objective that the shapes are respected even if the lengths are not. This is how Mercator does it. He decides to change the spacing between the parallels on his map. In each zone, it expands in the direction of the meridians in the same proportion as the parallels. Certainly, we lose the fact that the map was exact in the direction of the meridians, but at least the small disks on Earth are represented by disks on the map (of a different radius).
In the Mercator map, the meridians are represented by verticals, the parallels by horizontal lines, but the spacing between two parallels was no longer proportional to the difference in latitudes, as in the flat square. You can also verify that it is proportional to the inverse of the cosine of the latitude.
This map has been incredibly successful for a variety of reasons. The main one is that it was useful for sailors: a ship that is heading at a constant heading on Earth describes a curve (whose barbaric mathematical name is rhumb line) which is represented on the map by a straight line. Easy to trace the route. Mathematical advantage: this type of cards, which may not preserve the lengths, but which preserves the shapes of the infinitely small domains, is defined, without much imagination, as conforming. This concept has become fundamental in mathematics: passing from Latin to Greek, we are talking today about holomorphic functions. Their applications are innumerable, and we will talk about them (no doubt) in other briefs!
Today, when we give the formula that describes the Mercator map, we use logarithms. But in Mercator’s time, these logarithms had not yet been discovered! Besides, for a contemporary mathematician, the Mercator map is nothing other than the logarithmic function of a complex number. Mercator would not have recognized his card, he who ignored logarithms as much as complex numbers!
