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; the meaning "sum necessary to balance the two sides of an account" is from 1620s. The meaning "what remains or is left over" is by 1788, originally in commercial slang. The sense of "physical equipoise" is from 1660s; that of "general harmony between parts" is from 1732.
Many figurative uses are from the 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 beam, "living tree," but by late 10c. also "rafter, post, ship's timber," from Proto-Germanic *baumaz "tree" (source also of Old Frisian bam "tree, gallows, beam," Middle Dutch boom, Old High German boum, German Baum "tree," and perhaps also (with unexplained sound changes) Old Norse baðmr, Gothic bagms). This is of uncertain etymology (according to Boutkan probably a substrate word). The shift from *-au- to -ea- is regular in Old English.
The meaning "ray of light" developed in Old English, probably because beam was used by Bede to render Latin columna (lucis), the Biblical "pillar of fire." The meaning "directed flow of radiation" is from 1906. To be on the beam "going in the right direction" (1941) originally was an aviator's term for "to follow the course indicated by a radio beam."
The nautical sense of "one of the horizontal transverse timbers holding a ship together" is from early 13c., hence "greatest breadth of a ship," and slang broad in the beam, by 1894 of ships; of persons, "wide-hipped," by 1938.
"emit rays of light," c. 1400, from beam (n.) in the "ray of light" sense. The sense of "shine radiantly" is from 1630s; that of "smile radiantly" is from 1804; that of "to direct radio transmissions" is from 1927. Related: Beamed; beaming.