One of Somerville’s first interests, and her motivation to study Euclid’s Elements (which covers many aspects of geometry), was in navigation. You’re probably familiar with what mathematicians call Euclidean, or planar, geometry. This is the study of shapes, distances, and angles on a flat surface, such as a table. From a few underlying assumptions, postulates, or axioms, mathematicians can derive a huge number of results in geometry, including that the sum of the angles in a triangle is always 180°. This is a special property for triangles drawn on flat surfaces. However, the Earth is not flat, and so to understand navigation, one needs to describe how geometry works on a sphere (which is a good approximation to the shape of the Earth). It may surprise you to discover that the sum of the angles of any triangle drawn on a sphere is more than 180° - try it by drawing on a ball and measuring the angles! The smaller the triangle, the closer the sum of the angles is to 180°. This is because on small scales a sphere is almost flat. A consequence of the differences between spherical and Euclidean geometry is the challenge of producing a flat map of the Earth. There are many ways to do this and, ideally, a map would preserve properties such as area, shape, distance, and direction. However, due to the differences between spherical and Euclidean geometry, it is not possible to preserve all desired properties at once. For example, the Mercator projection, which corresponds to the standard map of the Earth, exaggerates the size of countries near the pole; Africa is a lot larger than the map makes you think it is!