A correspondent writes to ask if the US English time expression quarter of, as in quarter of four, is an ellipsis of something like 'It lacks/wants a quarter of four'.
I don't think it's necessary to suggest an implied verb. The preposition of has several locative uses, and it's a natural semantic extension to move from space to time. The original meaning of the preposition was 'away from' (a sense today now usually found with off), as seen in such obsolete usages as not far of the town and still found in relation to compass points (eg north of London) and specified distances (eg within a mile of).
In fact the OED (see of 4c) locates the clock sense along with other senses 'expressing position which is (or is treated as) the result of departure, and is defined with reference to the starting point', The time quarter of four, from this perspective, means 'a quarter away from four'.
The British use is quarter to, and is quite old. The OED (see to, 6b) has citations from around 1000 illustrating a wide range of usage, such as half hour to five and two hours to day (i.e. 'until daybreak'). The usage with of is less easy to track, because it has also had a dialect use in Scotland and Northern Ireland which probably antedates the US use, though citations are lacking. The earliest OED citation for the American use is 1817.