Every flat map of a round planet is wrong. The only question is which kind of wrong you are willing to accept, and Gerardus Mercator answered that question in 1569 in a way that has irritated people ever since and that nobody has been able to replace for the job he designed it for.
His map makes Greenland look roughly the size of Africa. Africa is about fourteen times larger. That is not an error. It is the direct and unavoidable consequence of the property that made the map useful, and Mercator knew exactly what he was trading away.
The Problem With Sailing in a Straight Line
A ship at sea in the sixteenth century navigated with a compass. The practical question for a pilot was: what single bearing do I hold to get from here to there?
A course held at a constant compass bearing is called a rhumb line, or loxodrome. On a globe it is not a straight line and not the shortest route. It is a spiral that crosses every meridian at the same angle and, if you follow it far enough, coils in towards the pole without ever arriving.
On the charts available before 1569 this curve was a curve, and an awkward one. Plane charts drawn on a simple grid of squares gave bearings that were wrong by amounts that grew with latitude, which is a serious matter when landfall depends on it.
Mercator’s insight was to ask what a map would have to look like for every rhumb line to be a straight line, and then build that map.
How the Stretch Works
Start by drawing the meridians as parallel vertical lines, evenly spaced. On the globe they converge at the poles, so this already stretches the map east to west, and the amount of stretching grows with latitude. At 60 degrees north the meridians have been pulled apart by a factor of two, and at 80 degrees by about six.
A bearing is an angle between a course and a meridian, so preserving angles means the map has to be stretched north to south by exactly the same factor at every point. Stretch equally in both directions locally and small shapes keep their form. That property is called conformality, and it is what makes the map work for navigation.
The consequence is unavoidable. Areas grow with the square of that factor, so at 60 degrees north an island is drawn four times too large, and near the poles the exaggeration runs away entirely. Mercator’s map cannot show the poles at all, since reaching 90 degrees requires infinite height.
Greenland sits between 60 and 83 degrees north. Africa straddles the equator. That is the whole explanation of the famous comparison.
Mercator Did Not Show His Working
The 1569 map, titled New and More Complete Representation of the Terrestrial Globe Properly Adapted for Use in Navigation, was printed on eighteen sheets and covered about two metres by one and a quarter. Mercator explained in a legend what the map was for and did not explain how he had constructed the spacing of the parallels. He appears to have worked it out geometrically, using a globe and a set of rhumb lines drawn on it, rather than by any formula.
The mathematics was supplied thirty years later by an Englishman, Edward Wright, in Certaine Errors in Navigation of 1599. Wright showed that the distance up the map to a given latitude is the running sum of the secant of the latitude, and he published a table of those sums computed at one minute intervals, which is a very large amount of arithmetic done by hand.
That the sum in question equals the logarithm of the tangent of half the co latitude was noticed by Henry Bond around 1645 as an empirical coincidence and proved by James Gregory and others in the 1660s. It is one of the first integrals in the modern sense to be evaluated in closed form, and it came out of a mapmaking problem rather than a mathematical one.
What Else He Did
Mercator was a Flemish engraver, instrument maker and globe maker who nearly did not survive his own century. In 1544 he was arrested for heresy along with several dozen others, held for seven months, and released only after the intervention of the University of Leuven. Two of the group were burned and one was buried alive.
He moved to Duisburg in the Duchy of Cleves in 1552, which was safer, and did most of his important work there. He also gave the word atlas to the language, using it in 1595 for his collection of maps in reference to the mythological king of Mauretania, an astronomer, rather than the Titan holding up the sky. The distinction mattered to him and has been lost ever since.
The Argument About What Maps Say
The complaint that the projection is politically loaded is a twentieth century one. In 1973 a German historian named Arno Peters presented, as his own invention, an equal area projection in which every country occupies its correct relative area, and argued that Mercator’s map had promoted a Eurocentric worldview by inflating the northern latitudes.
The projection he presented had been published by James Gall in 1855. Its shapes are severely stretched, and cartographers reacted to the presentation with unusual hostility. By 1989 seven North American geographic organisations had issued a joint statement urging that rectangular world maps of any kind be abandoned for general use.
The substantive point survives the mess: a world map used for general reference is making a claim about relative importance, and area is the most obvious thing a reader takes from one. The reply is equally sound: no flat map preserves everything, equal area projections destroy shape, and choosing a projection means choosing what the map is for.
The projections used for reference atlases now, such as Robinson and Winkel Tripel, preserve nothing exactly and distort everything slightly, which is a reasonable compromise for a map whose purpose is to be looked at rather than measured.
Why It Is Still on Your Phone
Mercator won the twenty first century by accident. Web mapping services use a variant known as Web Mercator, and they use it for two reasons.
- It is conformal. When you zoom into a city, streets meet at the angles they meet at in reality, and buildings are not sheared. For a map you navigate at street level this matters far more than the relative size of continents
- It tiles cleanly. The whole world becomes a square that divides neatly into four, and each of those into four again, all the way down. That structure is what lets a map server deliver small image tiles at any zoom level without recomputing anything
Web Mercator also cheats slightly: it treats the Earth as a sphere for the projection while using ellipsoidal coordinates from GPS, which makes it not quite conformal and unsuitable for surveying. For finding a restaurant it is fine.
So a construction devised for sixteenth century pilots, whose defining feature is that it exaggerates high latitudes without limit, is now the most widely viewed image of the Earth in history, and it is viewed mostly at a zoom level where the distortion is invisible.
The Right Way to Read It
The useful habit is to ask what any given map preserves and what it sacrifices, because it always sacrifices something. Mercator preserved angle and sacrificed area, deliberately, for a specific profession with a specific problem. Halley’s eclipse map of 1715 sacrificed almost everything else to show one thing moving across a country. Snow’s map of Soho threw away topography entirely to show deaths clustered around a pump.
A map is an argument about what matters. Mercator’s argument was that a sailor needs to reach the harbour alive, and on that question he was right for four hundred years.
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