This is a projection that at one time was quite popular in atlases.
It is a projection that is very simple to construct with compasses and straightedge, and which approximates the characteristics of the equatorial case of the azimuthal equidistant projection. Thus, when a hemisphere in which a compromise between conformal and equal-area characteristics is desired is to be drawn, it was a natural choice. Today, with computers to do our work for us, projections designed for ease of manual drawing have, though, become less popular.
The construction is simple enough. The equator is divided uniformly into points corresponding to the different longitudes, and the meridians are circular arcs drawn between these points and the poles. The bounding circle and the central meridian of the hemisphere are divided uniformly into points corresponding to the different latitudes, and the parallels are circular arcs joining those points.