The popular Hammer-Aitoff projection is based on an idea by Aitoff which he originally applied to the equatorial case of the azimuthal equidistant projection, yielding this:
rather than to the equatorial case of Lambert's azimuthal equal-area projection.
The Winkel Tripel projection, which is a popular projection in many European atlases,
is the arithmetic mean of it and a Plate Carrée projection with 40 degrees latitude as its standard parallels.
Actually, this is merely Bartholemew's variant of the Winkel Tripel projection, I have since learned; the arccosine of two over pi, which is about 50.459776 degrees, was the standard parallel used in Winkel's original Tripel projection, as illustrated below: