According to the TAC "Index of John Drewry's Dance"s:

Rinty's Rant was reworked and renamed to Bothwell Brig. Thus, Bothwell Brig replaced Rinty's Rant.

The dance was devised in the 1970ies, but the actual year is unknown.

The instructions say it is "for Nancy Rintoul of Bellshill". Also that Bothwell Bridge, on the Clyde, is famous as the scene of an encounter between Monmouth and the Covenanters in 1679.