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.
A modification of this projection is referred to in The Study of Map Projections by Steers, apparently invented by the cartographer A. M. Nell, also known for the Nell-Hammer projection. In this, the central meridian and Equator have scales placed on them which are the arithmetic mean of the uniform ones of the globular projection and those of the Stereographic, in which the parallels ane meridians are legitimately circles.
One can view the Globular projection as an attempt to approximate the equatorial case of the Azimuthal Equidistant. Since Airy's minimum-error azimuthal involves a limited expansion of radial scale away from the center, one could also, but it would be rather more of a stretch, view this projection as an approximation to Airy's projection!