Cartographic Projection
every map is wrong, and by exactly how much
What is taught wrongly

Mercator against Peters

The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction.
14 min read 8 figures Purpose before propertyComputed, not quotedThe trade-off is forced

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.

How much each projection inflates a cell, by latitudeFive patches of the sphere, each 20° by 10°, and the factor by which each projection enlarges them relative to the equatorial one. An equal-area projection sits flat on 1. Mercator reaches 15.4× at 70°, which is the mechanism behind every complaint about the size of Greenland.equator23°45°60°70°1×MercatorEquirectangularMillerGall–Peterslatitude of the cell1× means treated fairlyrelative to the equator
Fig. 1 How much each projection enlarges a fixed patch of the sphere as it moves poleward, relative to the same patch at the equator. Mercator reaches fifteenfold at 70°. The equal-area projections sit flat on one, by construction and by measurement.

The half that is right

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

Mercator’s areal scale factor is sec⁡2φ\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 figure above, and it is exact.

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

Five identical cells on MercatorFive patches, each 20° of longitude by 10° of latitude. On the sphere the higher ones are genuinely smaller, because the meridians converge. On Mercator the cell at 70° comes out 15.4 times larger than the equatorial one relative to its true size.1.0×1.3×2.4×5.6×15.4×each cell is 20° × 10°drawn in Mercator
Fig. 2 The mechanism, drawn. Five patches, each 20° by 10°. On the sphere the higher ones are genuinely smaller because the meridians converge; on Mercator they grow instead, and the number in each is the exact inflation factor.

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.

Tissot's indicatrix across Gall–PetersA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Gall–Peters ω reaches 39°, and the areal factor reaches 1.0.dashed: an undistorted circledrawn in Gall–Peters
Fig. 3 Gall–Peters with its indicatrices. Every ellipse has the same area, which is what equal-area means and is confirmed to a part in 10¹¹. Not one of them is a circle away from 45°, and near the equator the shapes are stretched vertically by a factor of two.

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.

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.

What a standard parallel buysThree equal-area cylindrical projections differing only in where they are exact. Each has zero angular deformation at its own standard parallel and grows away from it in both directions. Choosing a standard parallel is choosing which latitudes to treat well, and there is no choice that treats them all well.0°20°40°60°80°exact at equatorexact at 30°exact at 45°latitudeangular deformationall three are equal-area
Fig. 4 Three equal-area cylindrical projections differing only in where they are exact. All three preserve area perfectly; all three distort shape; and each has zero angular deformation only at its own standard parallel. The choice is free, and it decides which latitudes look right.

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.

Angular deformation against latitude, four projectionsThe same quantity for mercator, gallPeters, behrmann, mollweide, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.0°20°40°60°80°MercatorGall–PetersBehrmannMollweidelatitudeangular deformationalong a meridian
Fig. 5 Angular deformation for Mercator, Gall–Peters, Behrmann and Mollweide. Mercator is flat on zero. All three equal-area projections rise steeply, and they differ from each other by as much as they differ from Mercator — which is the point about equal-area underdetermining a map.

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.

6 projections of the same sphereThe same graticule under gallPeters, mollweide, eckert4, hammer, sinusoidal, behrmann. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve.Gall–PetersMollweideEckert IVHammerSinusoidalBehrmannsame sphere, same graticuleno two agree
Fig. 6 Six equal-area projections. All preserve area exactly — the measured error is 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 the first.

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 same point at 30°, 20° under four projectionsOne circle on the sphere, four projections, four ellipses. A conformal projection keeps it circular and lets the area run; an equal-area projection keeps the area and lets the shape go. Nothing keeps both, and the dashed circle shows what keeping both would look like.Mercatorω 0°area 1.13×Gall–Petersω 32°area 1.00×Mollweideω 11°area 1.00×Robinsonω 5°area 0.85×dashed: undistortedscaled to fit
Fig. 7 One point at 20° north under four projections. Mercator keeps the circle round. Gall–Peters stretches it vertically — this is the tropical shape distortion the projection’s advocates did not discuss. Mollweide and Robinson are between.

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.

Construction against propertyEvery projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has.conformalequal-areacompromiseclaim failscylindricalMercatorMercatorLambertGall–PetersBehrmannEquirectangularMillerWebpseudocylindricalSinusoidalMollweideEckertRobinsonpseudoazimuthalHammerWinkelazimuthalStereographicLambertOrthographicGnomonicAzimuthalconicLambertAlbersrows: how it is builtcolumns: what it preserves
Fig. 8 The audit table. Mercator sits under conformal, Gall–Peters under equal-area, and the empty intersection is where the argument was implicitly asking for something to be.

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.

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=R2 Δλ (sin⁡φ2−sin⁡φ1)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. The cell figures have no such ambiguity, and the cell areas are verified by summing a full covering of the sphere and requiring 4π4\pi, which comes out exact to 7×10−157\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.

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.

Where this goes next

New YorkMadridsolid: shortest · dashed: constant bearing+3.0% for the rhumbdrawn in Mercator Why Mercator exists A ship can hold a compass bearing and cannot easily hold a great circle. Mercator is the answer to one question — what must a map do so that a constant bearing is a straight line — and it answers it exactly. 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured The trade-off is two lines Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument. EquirectangularMercatorMollweideSinusoidalRobinsonWinkel tripelsame sphere, same graticuleno two agree Which projection is best An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.