Interestingly, The Shire in Middle-Earth is divided into four parts, each called a Farthing.
The word farthing derives from the Old English word fēorðing, from fēorða, meaning "a fourth" (or a quarter).
At the time Tolkien wrote The Hobbit and the Lord Of The Rings, the county of Yorkshire in England was divided into three administrative areas know as Ridings (East, West, and North Ridings). The word riding is descended from late Old English þriðing or þriding as a loanword from Old Norse þriðjungr, meaning a third part.
þriðing would be pronounced "thriding" which, as time went by started to be spoken as "the riding". Hence the subdivision of Yorkshire into Ridings.
In early English thrid was used rather than third (and brid instead of bird).
(Wow! A bit of mathematics, to possibly interest @Drew Espinosa - and bit about how the Vikings influenced parts of the English language, which may interest @Timothy M...)