mid-15c. (implied in indissolubly), from Latin indissolubilis "indestructible, that cannot be dissolved," from in- "not, opposite of" (see in- (1)) + dissolubilis, from dis- + solubilis (see soluble). Related: Indissolubly.
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."
In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
late 14c., "capable of being dissolved," from Old French soluble "expungable, eradicable" (13c.), from Late Latin solubilis "that may be loosened or dissolved," from stem of Latin solvere "to loosen, dissolve," from PIE *se-lu-, from reflexive pronoun *s(w)e- (see idiom) + root *leu- "to loosen, divide, cut apart." Meaning "capable of being solved" is attested from 1705. Substances are soluble, not solvable; problems can be either.