If you think about it the original "O" gauge was 1/4" scale or 1/4" = 1foot, if you assume the original "HO" gauge was supposed to be Half O gauge then 1/8" scale or 1/8" = 1 foot does make sense from a historic perspective. Since HO scale was later changed to its current 1/87 ratio this is where the change becomes apparent .
Not really. O scale (and HO) originated in Europe. O was originally 1:43.5 so HO was half O. (These days 1:45, the most accurate O scale, is used in Europe although 1:43.5 is also used, the UK sticks to 1:43.5 or 1:43). In the USA 1:48 was adopted for O scale as a result of the ease of measurement in feet and inches. In those days models were not that accurate so the scale discrepencies were not seen as important. Some USA makers may have used 1/8 scale for ease of measurement, but as far as I am aware HO has always been 1:87.
Hamish