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balance (n.)

early 13c., "scales, apparatus for weighing by comparison of mass," from Old French balance "balance, scales for weighing" (12c.), also in figurative sense; from Medieval Latin bilancia, from Late Latin bilanx, from Latin (libra) bilanx "(scale) having two pans," possibly from Latin bis "twice" (from PIE root *dwo- "two") + lanx "dish, plate, scale of a balance," which is of uncertain origin.

The accounting sense "arithmetical difference between the two sides of an account" is from 1580s; meaning "sum necessary to balance the two sides of an account" is from 1620s. Meaning "what remains or is left over" is by 1788, originally in commercial slang. Sense of "physical equipoise" is from 1660s; the meaning "general harmony between parts" is from 1732.

Many figurative uses are from Middle English image of the scales in the hands of personified Justice, Fortune, Fate, etc.; thus in (the) balance "at risk, in jeopardy or danger" (c. 1300). Balance of power in the geopolitical sense "distribution of forces among nations so that one may not dominate another" is from 1701. Balance of trade "difference between the value of exports from a country and the value of imports into it" is from 1660s.

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balance (v.)
1570s, "be equal with," from balance (n.). 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.
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balance-sheet (n.)
"statement showing the state of credits and debits in a particular business," 1812, from balance (n.) in the accounting sense + sheet (n.1).
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balance-beam (n.)
1798 as the cross-piece of a scales, 1813 as a type of device on a drawbridge, canal-lock, etc., from balance (n.) + beam (n.). From 1893 as a type of gymnastics apparatus.
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balanced (adj.)
1590s, "in equilibrium," past-participle adjective from balance (v.). In reference to meal, diet, etc., by 1908.
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unbalance (v.)
1856, from un- (2) "reverse, opposite of" + balance (v.).
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well-balanced (adj.)
1620s, from well (adv.) + past participle of balance (v.).
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unbalanced (adj.)
1640s, from un- (1) "not" + past participle of balance (v.). Earliest use is in reference to the mind, judgment, etc. Of material things, it is recorded from 1732.
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counterbalance (v.)

"to weigh against with an equal weight; to serve as a counterpoise to," 1610s, from counter- + balance (v.), in reference to scales. Figurative use dates from 1630s. As a noun, from 1610s, "equal weight or power;" figuratively, "influence acting in opposition," from 1630s.

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