late 14c., resolven, "melt, dissolve, reduce to liquid; separate into component parts; alter, alter in form or nature by application of physical process," " intransitive sense from c. 1400; from Old French resolver or directly from Latin resolvere "to loosen, loose, unyoke, undo; explain; relax; set free; make void, dispel."
This is from re-, here perhaps intensive or meaning "back" (see re-), + solvere "to loosen, untie, release, explain," from PIE *se-lu-, from reflexive pronoun *s(w)e- (see idiom) + root *leu- "to loosen, divide, cut apart."
From the notion of "separate into components" comes the sense in optics (1785; see resolution). From the notion of "reduce by mental analysis into its basic forms" (late 14c.) comes the meaning "determine, decide upon" after analysis (1520s), hence "pass a resolution" (1580s); "decide, settle" a dispute, etc. (1610s). For sense evolution, compare resolute (adj.).
In Middle English also "vaporize a solid, condense a vapor into a liquid, etc.;" a mid-15c. document has Sche was resoluyd in-to terys where a later writer might have she dissolved in tears. Related: Resolved; resolving.
"determination, firmness or fixedness of purpose; a determination," 1590s, from resolve (v.). Meaning "a determination of a deliberative body" is from 1650s.