1856, from un- (2) "reverse, opposite of" + balance (v.).
Entries linking to unbalance
un- 2
prefix of reversal, deprivation, or removal (as in unhand, undo, unbutton), Old English on-, un-, from Proto-Germanic *andi- (source also of Old Saxon ant-, Old Norse and-, Dutch ont-, Old High German ant-, German ent-, Gothic and- "against"), from PIE *anti "facing opposite, near, in front of, before, against" (from PIE root *ant- "front, forehead," with derivatives meaning "in front of, before").
More or less confused with un- (1) through similarity in the notions of "negation" and "reversal;" an adjective such as unlocked might represent "not locked" (un- (1)) or the past tense of unlock (un- (2)).
balance v.
1570s, "be equal with," from balance (n.). The meaning "serve as a counterpoise to" is from 1590s; that of "bring or keep in equilibrium" is from 1630s; that of "keep oneself in equilibrium" is from 1833. Of accounts, "settle by paying what remains due," from 1580s. Related: Balanced; balancing.