The early maps of the globe were a conjoined effort on mapmakers part. They combined local maps that had been made throughout the centrues in order to form the whole picture. But they could only approximate the actual dimension and shape of the continents on Earth, and this left much distortion. It wasn't until the 18th and 19th century that mapmakers began to truly understand how the continents were set on the Earth's surface.

It was through the science of Topology that scientists could determine almost precisely how the continents should appear on Earth. Topology is derived from Geometric Set Theory, which is now the foundation for Mathematical Analysis, Topology, Abstract Algebra, and Discrete Mathematics among other disiplines. Through Set Theory, Topology thus concludes that spatial properties are preserved under continous deformations--or--to put it simply, Geometry can calculate the way shapes will bend over time. Thus, mapmakers could calculate the shapes of Earh's land masses from the general measurements already obtained.

Sounds simple, but it wasn't until a few hundred years agon that guys like Giovanni Cassini (i.e Cassio watch) and Henri Poincare (perhaps the most underrated scientist/mathematician ever) began using and proving topology and more specificially, Algebraic Topology.

The proof is perhaps more easy to explain through the spatial representations of Geomety in the picture below than through the quantitative proofs of Set Theory.