Maps and Projections

The Theorem Underneath

This is not a complaint about cartographers. It is a theorem: a sphere cannot be flattened onto a plane without distortion. Gauss proved the general form of it — curved and flat surfaces have different intrinsic geometry, and no amount of cleverness gets around it.

So every projection chooses which property to save and which to sacrifice:

There is no neutral map. Not "we haven't found one yet" — there cannot be one. Which means the choice of projection is always an argument, and the only question is whether it was made deliberately.

Mercator Was Right (For His Job)

Mercator's 1569 projection is routinely denounced, usually by people who have not asked what it was for. It has a genuinely beautiful property: a line of constant compass bearing is a straight line on the map. For a sixteenth-century navigator that is not a convenience, it is the entire problem solved — you draw a straight line, read the bearing, and hold it.

The cost is that area inflates with latitude, and near the poles it inflates without bound. Greenland ends up looking the size of Africa when Africa is about fourteen times larger.

So Mercator is not a bad map. It is a superb navigational chart that spent four centuries hanging on classroom walls, doing a job it was never designed for. The scandal is not the projection; it is the promotion of a specialized tool to the status of "what the world looks like" — and then the fact that the tool systematically flatters high latitudes, which is where the people printing the maps happened to live.

Peters (1973) preserves area and mangles shape, and was marketed explicitly as a political corrective. Cartographers largely disliked it — its shapes are genuinely poor, and equal-area projections had existed for centuries without the sales pitch. But the argument it started was the right one, and it worked: nobody puts Mercator on a classroom wall innocently any more.

The Rest of the Arguments

Projection is only the mathematical layer. The rhetorical ones are simpler and often louder:

• Centring. Nothing is at the centre of a sphere's surface. Whatever you put there is a choice, and it reads as the middle of the world.
• Orientation. North-up is a convention, not a fact. Medieval T-O maps put east at the top (hence "orientation" — toward the orient); Islamic maps often put south up.
• Naming. Whose name, in whose language. A place with two names on a map has a border running through it.
• Line weight. A firm border and a dotted one are two claims about the same disputed line.
• Blankness. The famous one. Colonial maps' empty interiors were full of people — the blank was an assertion that nothing there merited recording, and an invitation to whoever arrived with the map.

Maps as Instruments of Power

Mapping is not describing; it is often the act by which a claim becomes administrable. The cadastral survey exists to tax land, and it must first make land legible: fixed boundaries, unique owners, measured areas — replacing tangles of customary, overlapping, seasonal rights that no column could hold. Once the survey is done, the customary rights are not merely unrecorded. They are unrecognized, because the register has become the reality the state can see.

The colonial version is starker. Borders drawn through territories the mapmakers had never visited became the borders of states, and remain so. The map preceded the reality, and then produced it.

The historian's discipline is therefore to read a map as a source, not a backdrop: who made it, for whom, what does it save and sacrifice, what is at the centre, what is named, what is blank — and above all, what did it make possible that was not possible before it was drawn.

What the Projection Buys, and What It Charges

The animation begins with the theorem, shown rather than stated: a globe's graticule is peeled and pressed toward a flat page, and it tears. It tears again, differently, at every attempt. The caption is short — a sphere has no flat copy, so every map is a bargain.

Then Mercator's bargain gets built. The cylinder wraps the globe; the parallels stretch as they climb; and crucially the animation shows why the stretch is not arbitrary — to keep angles true, the vertical stretch must match the horizontal one at every latitude, so the map buys conformality by paying in area, and the bill grows with latitude.

The payoff is demonstrated where it deserves to be. A navigator's rhumb line — constant compass bearing — curves absurdly across the globe, then snaps into a perfectly straight line on the Mercator sheet. This is the sixteenth century's hardest problem, solved. The animation lets that land before it collects.

Then the bill arrives. A Greenland-shaped tile lifts off the map and slides down to the equator, shrinking as it descends, until it settles beside Africa at its true size:

$$\text{Africa} \approx 14 \times \text{Greenland}$$

The final movement is the argument. The Mercator sheet is re-framed as what it is — a navigational chart, superb at its job — and then a classroom wall assembles around it, four centuries of it, flattering exactly the latitudes where the maps were printed. The closing caption does not blame the projection: the scandal was promoting a specialized instrument to "what the world looks like", and forgetting there was ever a bargain at all.