Mercator against Peters

In 1973 Arno Peters held a press conference in Bonn to announce a new world map. He argued that Mercator systematically enlarged the temperate latitudes where the colonial powers sat and shrank the tropics where their colonies were, and that this had shaped how generations understood the world.

Fifty years later the argument is still going, and it is conducted almost entirely without anybody measuring anything. Both quantities at issue are exactly computable.

The half that is right #

Peters was correct about the arithmetic, and the correction is not marginal.

Mercator’s areal scale factor is $\sec^2\varphi$. At 60° that is 4; at 70° it is about 8.5; at 80° about 33. Measured as the inflation of a fixed 20°-by-10° cell relative to the same cell at the equator, the site’s machinery gives 15.4× at 70°, computed from closed-form cell areas with no coastline dataset involved.

The consequence everyone quotes is Greenland against Africa. Africa is roughly fourteen times Greenland’s area; on Mercator they appear comparable. That comparison is a measurement of two real places and this site quotes it as one, but the mechanism is the cell arithmetic above, and that is exact.

So the claim that Mercator massively enlarges high latitudes relative to low ones is true, computable, and larger than most people expect.

The mechanism is worth stating rather than drawing twice. Five patches of 20° by 10°: on the sphere the higher ones are genuinely smaller because the meridians converge, and on Mercator they grow instead — the inflation factor in each is exactly the areal scale factor at that latitude.

The half that is not #

Peters presented the projection as the honest map, correcting a distortion rather than trading one for another. That part does not survive measurement.

Gall-Peters is equal-area to a part in 10¹¹, and not one of its indicatrices is a circle away from 45°: near the equator the shapes are stretched vertically by a factor of two. That is the half of the argument its advocates did not make.

The measured maximum angular deformation on Gall–Peters is about 146°. Shapes near the equator are stretched vertically and shapes near the poles are flattened horizontally, and the effect on the outlines of countries is severe — Africa, the continent the projection was advocated for, comes out noticeably elongated.

This is not a defect of Gall–Peters specifically. It is forced: conformal means the principal scales are equal, equal-area means their product is one, both together would be an isometry, and no isometry exists. An equal-area projection must distort shape somewhere, and the only question is where and how much.

Where and how much, in closed form #

Gall–Peters is simple enough that the answer can be written down rather than sampled, which makes it a useful check on the sampling.

The projection is $x = \lambda\cos 45°$ and $y = \sin\varphi / \cos 45°$, so the two scale factors are $h = \sqrt{2}\cos\varphi$ along the meridian and $k = 1/(\sqrt{2}\cos\varphi)$ along the parallel. Their product is exactly one at every latitude, which is the equal-area property with no measurement required. Their ratio is not:

$$\frac{h}{k} = 2\cos^2\varphi$$

At the equator that is exactly 2. A small circle on the ground comes out twice as tall as it is wide — not approximately, exactly. At 45° the ratio is 1 and the projection is locally faithful. At 70° it is 0.234, so a shape there is flattened by more than four to one in the other direction.

The angular deformation follows from the same two numbers, since the principal scales of a cylindrical projection are $h$ and $k$ themselves: $\omega = 2\arcsin\frac{a-b}{a+b} = 2\arcsin\frac13 = 38.94°$ at the equator. The site’s sampled machinery, which knows none of this algebra, returns 38.94° there and 146.4° at its worst.

The exactly-twofold vertical stretch falls on the equator, which is to say on the tropics — the latitudes the projection was advocated for.

So Gall–Peters is exactly as honest about area as Mercator is about angle, and exactly as dishonest about the other thing. Neither is the honest map, because there is no such object.

What “honest” would have to mean #

The word is doing the work in the argument, and it will not bear the weight.

A projection cannot be honest or dishonest, because it cannot make a claim. It is a function. What can be honest is a use of a projection: choosing one whose preserved property matches what the map is for, and saying which property that is.

By that standard:

• Using Mercator for navigation is honest, and it is what the projection was built for.
• Using Mercator for a classroom world map is not, because nothing about the classroom’s purpose has to do with compass bearings and the areal error is enormous.
• Using Gall–Peters for a thematic map of population or disease is honest, because those quantities are per unit area.
• Presenting Gall–Peters as the map without distortion is not, because its shape distortion is severe and measurable.

The argument, restated that way, is not about geometry at all. It is about what world maps are for, which is a real and worthwhile question that neither side was having.

The part of the argument nobody mentions #

Two further points, both awkward for the Peters case and neither usually raised.

It was not new. The projection is essentially the one James Gall published in 1855, which is why cartographers now call it Gall–Peters. Peters presented it as an original construction and appears to have been unaware of Gall; the cartographic establishment’s reaction was not improved by this.

The choice of standard parallel is arbitrary. Gall–Peters is exact at 45°. There is no reason it has to be.

Behrmann’s uses 30°, Lambert’s uses the equator, and there is a continuum between them. Every one is equal-area. Choosing 45° rather than 30° makes the temperate zones look better and the tropics worse in shape, which is an odd choice for a projection advocated on the grounds of fairness to the tropics — and it is a choice, not a consequence.

What the argument achieved #

Something worth having, in the end, and not what either side thought.

The cartographic establishment’s response in the 1970s and 1980s was largely to attack Peters’ competence and originality, which was accurate and beside the point. The substantive question — whether the default world map should be conformal — is a good one, and the answer that emerged was that both were poor defaults.

By 1989 seven North American geographic organisations had issued a joint resolution against using any rectangular world projection for general-purpose maps, Mercator and Gall–Peters alike. The National Geographic Society had already moved to Robinson in 1988 and to Winkel tripel in 1998, both compromises that preserve nothing exactly.

That is the right conclusion and it took a bad-tempered public argument to reach it.

What each side was actually claiming #

Separating the claims makes the disagreement tractable, and most accounts do not separate them.

Peters’ geometric claim: Mercator grossly enlarges high latitudes relative to low ones. True, and this site measures it at fifteenfold for a 70° cell against an equatorial one.

Peters’ historical claim: this was not accidental and served colonial interests. Not supported. Mercator’s construction follows from the navigational requirement and the distortion is a forced consequence of it. The projection predates the colonial period it is said to serve, and no alternative construction meeting the navigational specification would have distorted less.

Peters’ cartographic claim: his projection is therefore the honest one. False, in the sense that no projection can be honest, and its own shape distortion is severe.

Peters’ pedagogical claim: the default world map shapes how people understand global proportion. Untested, and probably untestable in the strong form, though the concern is reasonable.

The cartographic establishment attacked the third claim and the projection’s originality, largely ignored the first, and never really engaged the fourth. Peters largely ignored the objection to the third. That is why the argument ran for twenty years without converging.

The equal-area projections nobody argued about #

A curiosity of the controversy is that Gall–Peters is not an especially good equal-area projection, and better ones were available throughout.

Six equal-area projections in this site’s own library preserve area exactly — the measured error runs between 6×10⁻¹² and 2×10⁻¹¹ for every one of them — and they look nothing alike. The argument was conducted almost entirely about Gall–Peters.

Mollweide dates from 1805, Eckert IV from 1906, Hammer from 1892. All are equal-area, all handle shape better than Gall–Peters over most of the map, and all were well known.

Peters’ choice of the cylindrical form is what made the map recognisable and controversial: it looks like a familiar rectangular world map with the proportions changed, which made the comparison with Mercator immediate. A Mollweide would have made the same areal point with far less shape damage and would not have looked like a rebuttal.

That is a presentational choice doing more work than the geometry, and it is probably why the argument was so heated.

What a map user should take from it #

Three things, none of which requires taking a side.

Areal distortion is real, large, and invisible. It does not look like anything, which makes it the more dangerous of the two failures.

Neither projection is neutral, and neither is dishonest. Both are exactly what they claim; the choice between them is about purpose.

For a general-purpose world map, both are poor choices and a compromise is better. That is the conclusion the field actually reached, and it took the argument to reach it.

What a fair comparison looks like #

Setting the two projections side by side with their measurements attached is the comparison the argument never quite had.

Mercator: angular deformation zero everywhere, areal inflation reaching fifteenfold at 70° and unbounded at the poles.

Gall–Peters: areal error zero everywhere, angular deformation reaching about 146°.

Both statements are exact. Neither projection is failing at anything it claims. And the two failures are independent quantities with no exchange rate, so there is no arithmetic that converts one into the other and declares a winner.

What is left is the purpose, and the purposes differ: one was built for navigation and one for thematic mapping of area-based quantities. Neither was built to be a classroom world map, and both are poor at it.

The measurement that should have ended it #

There is a single figure that would have shortened the argument considerably, and it is the one this site’s machinery produces first.

Run both projections through both tests. Mercator passes conformality and fails equal-area. Gall–Peters passes equal-area and fails conformality. Neither passes both, and nothing ever will.

That table says everything the geometry has to contribute, in four cells, and it says it without adjudicating. The remaining disagreement — which failure a general-purpose map should accept — is real and is not geometric, and separating the two would have saved twenty years.

A final point about how the argument was conducted. Both sides had access to the same measurements, and neither made them the centre of the case. The geometry was settled throughout and the disagreement was about purposes, and presenting a purpose disagreement as a factual one is what kept it going — which is a general hazard whenever a technical vocabulary is available to dress up a preference.

The episode is also a case study in what a measurement does to an argument. Almost every claim either side made was checkable, none of the checking was done publicly, and the disagreement ran on assertion for twenty years. That is not a failure of the participants so much as a demonstration that having the tools available is not the same as anyone reaching for them.

Reading the episode now, the most striking thing is how much of it turned on a word. “Fair” and “honest” carried the argument on one side and “accurate” carried it on the other, and none of the three is a property a function can have. The measurable content was settled throughout; the vocabulary is what took twenty years.

The argument is usually conducted over the whole world, where both projections are at their worst. Over a band where both are usable, the difference between them changes character.

Across the tropics the two projections put their worst points in different places for a reason with a theorem behind it. Mercator’s is on the boundary because it is conformal; Gall-Peters has its worst point at the equator, in the centre of the band, because its standard parallels are outside the band altogether.

What was computed here #

Every number in this essay comes from the site’s own machinery.

Areas are computed from the closed form for a latitude–longitude cell, $A = R^2\,\Delta\lambda\,(\sin\varphi_2 - \sin\varphi_1)$, which is exact. The site deliberately does not use a coastline dataset for this: a polygon’s area depends on its simplification level, so a Greenland-against-Africa figure computed from Natural Earth would be partly a measurement of the vendor’s generalisation. A latitude–longitude cell has no such ambiguity, and the cell areas are verified by summing a full covering of the sphere and requiring $4\pi$, which comes out exact to $7\times10^{-15}$.

The angular deformations are measured from the projections’ own derivatives at several hundred points, taking the worst rather than the mean. Gall–Peters is asserted to pass the equal-area test and to fail the conformality test, and both assertions run on every build — the second is as important as the first, because an equal-area test that accepted everything would make this essay’s central comparison meaningless.

What the pictures cannot show #

The cells are 20° by 10° patches, not countries. That is deliberate and it is a limitation: a real country has a shape, spans a range of latitudes, and its apparent size on a map is not a single number. The cells give an exact answer to a slightly different question, and the trade is between exactness and directness.

The essay also cannot show what a map does to somebody’s understanding of the world, which is the claim Peters actually made. That a projection inflates high latitudes by fifteenfold is measurable; that this shaped a generation’s sense of global proportion is a claim about people, and no amount of geometry supports or refutes it.

What each side’s projection does to a computation #

The argument is usually about what a reader infers from the sizes of shapes. There is a sharper version, which is what a program computes from them, and it is not a matter of impression.

A shoelace over a mid-latitude cell returns 3.06 times the true area in the conformal plane and the closed-form answer to ten decimal places in the equal-area one. In the other direction a scale bar on the equal-area map reads 500 kilometres along the parallel and 2,000 along the meridian at 60° north — reciprocal, because that is what equal-area means — so the projection recommended for honest areas is the one on which no single correction repairs a distance.

As an operation the dispute has a clean answer and it is not the one either side argued for. The equal-area member returns a cell’s closed-form area exactly and the conformal one returns three times it, which is the same ratio the argument is about — computing an area needs a surface measures it rather than displaying it. The cost on the other side is just as measurable: a bar on the equal-area map reads short one way and long the other with the product exactly one at every latitude, while on the conformal map the two readings agree, so one number per latitude corrects any measurement taken off it.

One failure saturates and the other does not #

There is an asymmetry in the two complaints that explains a great deal about how the argument sounded, and it is a fact about the measures rather than about the debaters.

The areal failure is unbounded. On the conformal map the areal factor is $\sec^2\varphi$: 33 at 80°, 3,283 at 89°, and no ceiling. However bad it is, it can always be twice as bad further north, and the number stays a plain ratio that anybody can repeat — Greenland looks fourteen times too big is a sentence.

The angular failure saturates. The maximum angular deformation is $\omega = 2\arcsin\!\frac{a-b}{a+b}$, and as the axis ratio runs away $\omega$ climbs towards 180° and stops. On the equal-area map it is 39° at 60°, 125° at 80° and 172° at 89° — already within eight degrees of the ceiling, with the axis ratio still only 1,642 and rising.

So the two sides were quoting incommensurable scales. One could say seven times too large and be understood; the other had to say an angle wrong by nearly the largest amount an angle can be wrong by, which sounds like rhetoric and is a measurement whose units have run out.

The saturation is real and it is not a defence. A shape distorted by 172° is not nearly as wrong as one distorted by 179°; the measure has simply stopped distinguishing them, and the axis ratio behind it has not.

Who found it, and when #

James Gall described the orthographic cylindrical equal-area projection with standard parallels at 45° in 1855, in a paper to the British Association. It attracted no particular attention.

Arno Peters, a German historian and film-maker rather than a cartographer, announced his version in 1973. The controversy ran for two decades and was unusually personal on both sides.

Johann Heinrich Lambert had constructed the general cylindrical equal-area projection in 1772 — along with the conformal conic, the azimuthal equal-area and several others, in a single publication that also named the properties as the design goals. Most of the projections in this argument, on both sides, descend from work Lambert did two hundred years before it started.

Where this goes next #

What Mercator was actually built for is why Mercator exists. Why neither projection could have both properties is the trade-off is two lines. And for the question the argument should have been about, which projection is best.