1580s, "resolution of anything complex into simple elements" (opposite of synthesis), from Medieval Latin analysis (15c.), from Greek analysis "solution of a problem by analysis," literally "a breaking up, a loosening, releasing," noun of action from analyein "unloose, release, set free; to loose a ship from its moorings," in Aristotle, "to analyze," from ana "up, back, throughout" (see ana-) + lysis "a loosening," from lyein "to unfasten" (from PIE root *leu- "to loosen, divide, cut apart").
Meaning "statement presenting results of an analytic process" is from 1660s. Psychological sense is from 1890. English also formerly had a noun analyse (1630s), from French analyse, from Medieval Latin analysis. Phrase in the final (or last) analysis (1844), translates French en dernière analyse.