E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
To renounce acquaintance. There are four sorts of cut
(1) The cut direct is to stare an acquaintance in the face and pretend not to know him.
(2) The cut indirect, to look another way, and pretend not to see him.
(3) The cut sublime, to admire the top of some tall edifice or the clouds of heaven till the person cut has passed by.
(4) The cut infernal, to stoop and adjust your boots till the party has gone past.
There is a very remarkable Scripture illustration of the word cut, meaning to renounce: Jehovah took a staff and cut it asunder, in token that He would break His covenant with His people; and He cut another staff asunder, in token that He would break the brotherhood between Judah and Israel (Zech. xi. 714).