sleeve n.

Middle English sleve, from Old English sliefe (West Saxon), slefe (Mercian) "arm-covering part of a garment," probably literally "that into which the arm slips," from Proto-Germanic *slaubjon (source also of Middle Low German sloven "to dress carelessly," Old High German sloufen "to put on or off"), from PIE root *sleubh- "to slide, slip."

It is related etymologically to Old English slefan, sliefan "to slip on (clothes)" and slupan "to slip, glide." Also for the sense, compare slipper, Old English slefescoh "slipper," slip (n.2) "woman's garment," and expression slip into "dress in."

The mechanical sense of "tube in which a rod or another tube is inserted" is by 1864. The meaning "the English Channel" translates French La Manche, literally "the sleeve" (from Old French manche "a sleeve," also "a handle," from Latin manicae "long sleeves of a tunic;" see manacle (n.)).

The figurative expression have something up (or in) one's sleeve, "have ready as occasion demands," is recorded from c. 1500 (the long, pendant sleeves of the late Middle Ages also sometimes doubled as pockets); to have a card (or ace) up one's sleeve in the figurative sense of "have a hidden resource" is from 1863; the cheat itself is mentioned by 1840s. To wear one's heart on (one's) sleeve is from "Othello" (1604). For laugh in one's sleeve see laugh (v.).