The Stereographic projection, particularly on hemispheres as shown here:
was very common on antique maps of the world.
This is a map projection that was known to the ancient Greeks. Like the Mercator projection, it is conformal. In addition to being used for maps, it is also sometimes used in crystallography.
It has the interesting and unusual property that all circles on the globe, small or great, appear as circles on the projection. This simplifies drawing it.
The Stereographic is constructed by projecting the globe onto a tangent plane from a "light source" located at the point on the globe antipodal to the plane's point of tangency - the spot where the plane on which one is drawing the map touches the globe.
The following diagram:
illustrates this principle.