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Old English beam originally "living tree," but by late 10c. also "rafter, post, ship's timber," from Proto-Germanic *baumaz (cf. Old Norse baðmr, Old Frisian bam "tree, gallows, beam," Middle Dutch boom, Old High German boum, German Baum "tree," Gothic bagms), perhaps from PIE verb root *bheue- "to grow" (see be). The shift from *-au- to -ea- is regular in Old English.
Meaning "ray of light" developed in Old English, probably because it was used by Bede to render Latin columna lucis, the Biblical "pillar of fire." 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 "wide-hipped" (of persons). To be on the beam (1941) was originally an aviator's term for "to follow the course indicated by a radio beam."
"emit rays of light," early 15c., from beam (n.) in the "ray of light" sense. Sense of "to smile radiantly" is from 1804; that of "to direct radio transmissions" is from 1927. Related: Beamed; beaming.
occurs in the Authorized Version as the rendering of various Hebrew words. In 1 Sam. 17:7, it means a weaver's frame or principal beam; in Hab. 2:11, a crossbeam or girder; 2 Kings 6:2, 5, a cross-piece or rafter of a house; 1 Kings 7:6, an architectural ornament as a projecting step or moulding; Ezek. 41:25, a thick plank. In the New Testament the word occurs only in Matt. 7:3, 4, 5, and Luke 6:41, 42, where it means (Gr. dokos) a large piece of wood used for building purposes, as contrasted with "mote" (Gr. karphos), a small piece or mere splinter. "Mote" and "beam" became proverbial for little and great faults.