The original Van der Grinten projection was, and continues to be, a very popular projection for world maps:
However, the fact that it distorts areas in a fashion reminiscent of the Mercator projection, if not quite as severly, is understandably perceived as a problem.
The Van der Grinten III projection deals with this by making the parallels straight lines, at the same heights as the parallels intersect the central meridian in the original Van der Grinten projection.
The resulting projection:
is still not a pseudocylindrical projection, because the meridians are circular arcs, thus being the same as in the original Van der Grinten projection.
Of course, usually the top and bottom of the projection will be trimmed off: