"slow, graceful dance in triple measure," 1670s, from French menuet, from Old French menuet (adj.) "small, fine, delicate, narrow," from menu "small," from Latin minutus "small, minute" (from PIE root *mei- (2) "small"). So called from the short steps taken in the dance. The spelling was influenced in English by Italian minuetto. As "music for a minuet," by 1680s. Perhaps invented in mid-17c. France, the minuet was, through the 18c., the most popular of the more stately dances.
Proto-Indo-European root meaning "small."
It forms all or part of: administer; administration; comminute; diminish; meiosis; Menshevik; menu; metier; mince; minestrone; minim; minimum; minister; ministration; ministry; minor; minuend; minuet; minus; minuscule; minute; minutia; Miocene; mis- (2); mite (n.2) "little bit;" mystery (n.2) "handicraft, trade, art;" nimiety.
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit miyate "diminishes, declines;" Greek meion "less, smaller;" Latin minus, minor "smaller," minuere "to diminish, reduce, lessen;" Old English minsian "to diminish;" Russian men'she "less."
For as her natural face decays, her skill improves in making the artificial one. Well, nothing diverts me more than one of those fine, old, dressy things, who thinks to conceal her age by everywhere exposing her person; sticking herself up in the front of a side-box; trailing through a minuet at Almack's; and then, in the public gardens looking, for all the world, like one of the painted ruins of the place. [Goldsmith, "The Good Natured Man," 1768]