©Fernando Caracena 2017
The above title is rather redundant. Geometry literally means "earth measure"; but as first developed by Euclid of Alexandria, its focus was on the descriptions of geometrical shapes that could be accommodated to a flat surface, such as a sheet of paper. The Earth certainly seems flat to us small creatures that crawl like ants over its surface, but as humanities activities have increased over its surface, through the school of hard knocks people have learned that Euclid' s description of geometrical relationships over the Earth's surface was too limited and therefore required somegeneralized.
Spherical geometry
As humans explored more of the Earth, in the process of enlarging their knowledge of distant lands and widening their trades, they found that the geometry of Euclid was inadequate to describe the relationships of large geometrical figures over the surface of the Earth. Some of the great thinkers among the Greeks believed that the Earth was a sphere and not flat. In Alexandria, Egypt, the Greek scholar and philosopher, Eratosthenes (276 BC – 195 BC), came close to measuring the radius of the Earth based on the angles of shadows cast by the sun in Egypt on the Summer solstice. The model that he used for this purpose was that of a spherical earth embedded in a flat, three dimensional space.
The idea that the Earth is nearly spherical was forgotten somehow in ancient times, but was regained in more modern times. In exploring large portions of the surface of our planet, map makers became aware that you just could not faithfully represent large portions of that surface on a flat sheet of paper without distortions. The way to get the geometry right was to draw it on a sphere. In those days, the map of the Earth on globes in someone's study became the mark of an well educated person.
Confronted by the reality of living on an almost spherical planet, mathematicians were almost forced to generalize Euclid' s plain geometry to that of a spherically curved space. The two dimensional surface of the planet Earth is not flat. It is a curved surface, which is very nearly a spherical surface. Although horizontal surfaces on the Earth seem quite flat to us, they are really small sections of spherical surfaces, which to us seem quite flat. Mathematically, the surface of a sphere can be described as a two dimensional manifold.
There are direct confirmations that the Earth has a spheroidal shape. The Earth's shadow cast on the moon during a lunar eclipse shows that its edge is circular. One of Galileo's early observations using a telescope was that sailing ships coming into harbor from far away were partially hidden from sight by the curvature of the earth's blocking light, which travelled very close to a straight line to the observer's eye. On his home-made telescope Galileo could see a the image of a boat approaching harbor, first only as the top of its sails, and progressively through the rest of it rising above the horizon.
The geometry of a two dimensional spherical surface is quite different from that of a flat plain. The curved space space of the earth's surface had to be handled a special way, as was discovered by the early map makers.
For convenience, navigators wanted maps that they could lay out flat on their desks. Unfortunately, reality did not allow the map makers to make flat maps of the earth without stretching or tearing the surface. They used map projections, which gave fairly accurate representations of the earth's surface on a flat map by allowing them to have systematic distortions. It became even possible to do accurate calculations on a flat map of a curved surface, by treating the distortions as functions of the map coordinates, which are called map scale factors.