O gauge is 1.25 inches and O scale is 1/4" = 1 ft. But that makes gauge 5 ft in O scale. So to rationalize the scale and gauge they used 7 mm = 1' scale to get proper standard gauge (well actually
4' 6
1/2") with commercial track and wheelsets.
HO was then "half O" or 3.5 mm = 1 ft.
Then the HO'ers decided to make gauge a scale 4'8 1/2", actual standard gauge, so even though its "half O", HO gauge (.649 in)
isn't
half of O gauge (.625 in).
Then later some of the O gaugers decided to use 1/4" scale and
make
scale standard gauge track so they use 1/4" = 1 ft O scale and
1.177
in gauge, which was at one time proposed to be called called Q