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.
Old English sciete (West Saxon), scete (Mercian) "cloth, covering, towel, shroud," from Proto-Germanic *skautjon-, from *skauta- "project" (source also of Old Norse skaut, Gothic skauts "seam, hem of a garment;" Dutch schoot; German Schoß "bosom, lap"), from PIE root *skeud- "to shoot, chase, throw."
Sense of "piece of paper" first recorded c. 1500; that of "any broad, flat surface" (of metal, open water, etc.) is from 1590s. Of falling rain from 1690s. Meaning "a newspaper" is first recorded 1749. Sheet lightning is attested from 1794; sheet music is from 1857. Between the sheets "in bed" (usually with sexual overtones) is attested from 1590s; to be white as a sheet is from 1751. The first element in sheet-anchor (late 15c.) appears to be a different word, of unknown origin.