For thousands of years, people have tried to show the surface of the Earth on a flat sheet of paper. This task sounds simple, but it contains a basic geometric problem. The Earth is roughly a sphere, and a curved surface cannot be flattened without some stretching or tearing. A useful comparison is the peel of an orange. If someone removes the peel and presses it onto a table, it will split or crease. Mapmakers face the same difficulty. The mathematical method they use to move points from a globe to a flat surface is called a projection, and every projection changes the shape of the world in some way.
The earliest known attempts to solve this problem came from the ancient Mediterranean world. Greek scholars already understood that the Earth was round, and some of them drew grids of lines to organise their maps. In the second century CE, the geographer Ptolemy described two methods for drawing the known world. In both of them, the lines of latitude were drawn as curved arcs rather than straight lines. This was an early form of what is now called a conic projection, because it imagines a cone placed over the globe. However, Ptolemy's work was largely forgotten in Western Europe for many centuries. It only became widely studied there again around 1400, when Greek manuscripts were translated into Latin.
The most famous projection was introduced in 1569 by the Flemish mapmaker Gerardus Mercator. His map was designed for a practical purpose: sea navigation. On a Mercator map, the lines of longitude are straight and parallel, and they meet the lines of latitude at right angles. As a result, a sailor can draw a straight line between two ports and follow a single compass direction for the whole journey. This kind of route is known as a rhumb line. Mercator did not publish a full mathematical explanation of his method. In fact, the exact formula behind his projection was worked out by other scholars several decades later.
Mercator's design kept the shapes of small areas correct, but it created a serious problem with size. In order to keep angles accurate, the projection stretches regions more and more as they move away from the equator. For example, Greenland appears roughly as large as Africa on a Mercator map. In reality, Africa is about fourteen times bigger. Countries far from the equator, such as Canada and Russia, also look much larger than they are. The poles themselves cannot be shown at all, because they would have to be stretched to an infinite size. For this reason, most printed Mercator maps simply cut off the far north and far south.
Other mapmakers chose a different priority. Instead of protecting shapes, they tried to keep the areas of regions in the correct proportion. These are called equal-area projections. In the late eighteenth century, the Swiss mathematician Johann Heinrich Lambert developed several new projections, including some equal-area designs that are still used today. On an equal-area map, a country that is twice as large as another will also cover twice as much space on the paper. However, this accuracy has a cost. Shapes on such maps are often squeezed or stretched, especially near the edges, so familiar coastlines can look strangely flattened. Mathematicians later proved that no flat map can preserve both true shape and true area at the same time.
Because every projection has weaknesses, many twentieth-century cartographers looked for a middle path. They designed compromise projections, which do not keep any single property perfectly but reduce the most obvious errors overall. One well-known example is the Robinson projection, created in 1963. Its designer did not begin with a mathematical formula. Instead, he adjusted the map by eye until the continents looked balanced and natural. Another compromise design, the Winkel tripel projection, was proposed in 1921 and later became popular for world maps in atlases. To help compare such designs, scholars use a tool invented in the nineteenth century by the French mathematician Nicolas Auguste Tissot. It places small circles across a map and shows how each circle is stretched into an oval by the projection.
Today, the choice of projection is shaped strongly by digital technology. Most online map services use a version of the Mercator projection. This may seem surprising, given its size problems. However, it has an important advantage for web maps: when users zoom in on a street or a building, the shapes and angles remain correct. In addition, the map can easily be divided into square tiles that load quickly on a screen. For maps of the whole world, on the other hand, many teachers and publishers now prefer compromise or equal-area designs. The long history of map projections therefore shows that there is no single correct map. Each method is a choice, and each choice depends on what the map is meant to do.