The county of Yorkshire was traditionally split into "ridings" which some still like to explain as meaning a distance or area that could be ridden in a day… unlikely, since even the question of whether it should be distance or area is unanswerable and Yorkshire is too large and hilly for much of a Riding to be ridden in a day…
Another explanation is that Riding is a distortion, through whatever path, of "thirding", taken from the way the county was split…
In a wholly different context when two families or kingdoms with coats of arms are joined by marriage, the heralds often split their arms in four parts of the shield, filling top left and bottom right with smaller images of the one, top right and bottom left with the other. This is called "quartering".
Unless someone sees a significant difference in the derivation of "thirding" or "quartering", their existence would seem to indicate an algorithm that could be followed with any number without changing any grammatical rule but which almost certainly hasn't been developed for the simple reason that it would be too clumsy. "Fifthing" might not be too bad but "twenty-seventhing" prolly would.
The early use of "quartering" seems quite rightly to treat people and numbers the same way, as things, albeit the one is concrete and the other abstract. In any case, "quartering" a person did mean literally splitting the body into four pieces.
Hind- or fore-quarters, on people or pigs or whatever creature start to blur the point a lot. Strictly the left leg might be a hind-quarter; both legs should presumably be a hind-half but the expression is always plural "hind-quarters"…
Similarly, the "(choose-an-ethnic-group) quarter" of a city is very rarely really a quarter of anything and the same lack of definition is more evident both in describing a soldier's "quarters" and in the verb describing that soldier as being "quartered" anywhere…
