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"). Related to Old English slefan, sliefan "to slip on (clothes)" and slupan "to slip, glide," from PIE root *sleubh- "to slide, slip."
Compare slipper, Old English slefescoh "slipper," slip (n.2) "woman's garment," and expression slip into "dress in." Mechanical sense is attested from 1864. 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.)).
To have something up one's sleeve is recorded from c. 1500 (large sleeves formerly doubled as pockets); to have a card (or ace) up one's sleeve in the figurative sense "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).
in poetic use, "a swallow" ("But the poets appear to have thought it some song-bird" - OED), late 14c., also Progne, from Latin Procne, Progne, from Greek Proknē, in mythology the name of the daughter of Pandion and sister of Philomela, who was transformed into a swallow.
"state of great difficulty or perplexity," 1570s, a word of unknown origin and even the pronunciation is unsettled in old dictionaries (it seems to have been originally accented on the second syllable). Perhaps it is a quasi-Latinism based on Latin quando "when? at what time?; at the time that, inasmuch," pronominal adverb of time, related to qui "who" (from PIE root *kwo-, stem of relative and interrogative pronouns).
mid-15c., "neither more nor less than, with no error; exactly stated or marked off; definitely or strictly expressed; distinguished with precision from all others," from Old French précis "condensed, cut short" (14c.) and directly from Medieval Latin precisus, from Latin praecisus "abrupt, abridged, cut off," past participle of praecidere "to cut off, shorten," from prae "before" (see pre-) + -cidere, combining form of caedere "to cut" (from PIE root *kae-id- "to strike"). For the Latin vowel change, see acquisition. Related: Precisely (late 14c.).