Etymology
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beam (n.)

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.

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mere (adj.)

late 14c., of a voice, "pure, clear;" mid-15c., of abstract things, "absolute, sheer;" from Old French mier "pure" (of gold), "entire, total, complete," and directly from Latin merus "unmixed" (of wine), "pure; bare, naked;" figuratively "true, real, genuine," according to some sources probably originally "clear, bright," from PIE *mer- "to gleam, glimmer, sparkle" (source also of Old English amerian "to purify," Old Irish emer "not clear," Sanskrit maricih "ray, beam," Greek marmarein "to gleam, glimmer"). But de Vaan writes "there is no compelling reason to derive 'pure' from 'shining,'" and compares Hittite marri "just so, gratuitously," and suggests the source is a PIE *merH-o- "remaining, pure." 

The English sense of "nothing less than, in the fullest sense absolute" (mid-15c., surviving now only in vestiges such as mere folly) existed for centuries alongside the apparently opposite sense of "nothing more than" (1580s, as in a mere dream).

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band (n.1)

"a flat strip," also "something that binds," Middle English bende, from Old English bend "bond, fetter, shackle, chain, that by which someone or something is bound; ribbon, ornament, chaplet, crown," with later senses and spelling from cognate Old Norse band and technical senses from Old French bande "strip, edge, side" (12c., Old North French bende), all three ultimately from Proto-Germanic *bindan, from PIE root *bhendh- "to bind."

The meaning "a flat strip" (late 14c.) is from French. In Middle English, this was sometimes distinguished by the spelling bande, bonde, but with loss of terminal -e the words have fully merged via the notion of "flat strip of flexible material used to wind around something."

The meaning "broad stripe of color, ray of colored light" is from late 14c.; the electronics sense of "range of frequencies or wavelengths" is from 1922. Most of the figurative senses ("legal or moral commitment; captivity, imprisonment," etc.) have passed into bond (n.), which originally was a phonetic variant of this band. The Middle English form of the word is retained in heraldic bend (n.2) "broad diagonal stripe on a coat-of-arms."

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mermaid (n.)

"fabled marine or amphibian creature having the upper body in the form of a woman and the lower in the form of a fish, with human attributes," "usually working harm, with or without malignant intent, to mortals with whom she might be thrown into relation" [Century Dictionary]; mid-14c., meremayde, literally "maid of the sea," from Middle English mere "sea, lake" (see mere (n.1)) + maid.

Old English had equivalent merewif "water-witch" (see wife), meremenn "mermaid, siren" (compare Middle Dutch meer-minne, Old High German meri-min), which became Middle English mere-min (c. 1200) and was shortened to mere "siren, mermaid" (early 13c.); the later mermaid might be a re-expansion of this. Tail-less in northern Europe; the fishy form is a medieval influence from the classical siren, and mermaids sometimes were said to lure sailors to destruction with song.

A favorite sign of taverns and inns at least since early 15c. (in reference to the inn on Bread Street, Cheapside, London). Mermaid pie (1660s) was "a sucking pig baked whole in a crust." Mermaid's purse for "egg-case of a skate, ray, or shark" is by 1825, perhaps originally Scottish, as it is first attested in Jamieson.

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shaft (n.1)

"long, slender rod," originally "staff or pole forming the body of a spear or lance; spear-shaft," also, perhaps by synecdoche, "spear;" Middle English shafte, from Old English sceaft from Proto-Germanic *skaftaz (source also of Old Norse skapt, Old Saxon skaft, Old High German scaft, German schaft, Dutch schacht, not found in Gothic).

OED suggests this might be explained as a Germanic passive past participle of PIE root *(s)kep- "to cut, to scrape" (source of Old English scafan "to shave, scrape, polish") on notion of "tree branch stripped of its bark." But compare Latin scapus "shaft, stem, shank," Greek skeptron "a staff" (see scepter) which appear to be cognates.

Extended generally to any body of long, cylindrical shape; the meaning "beam or ray" (of light, etc.) is attested from c. 1300; that of "arrow" (especially a long one, used with a long bow) is from c. 1400; that of "a long, straight handle of a tool or utensil" from 1520s. The mechanical sense "long rotating rod for transmission of motive power in a machine" is from 1680s.

The vulgar slang meaning "penis" is recorded by 1719 on notion of "columnar part" (late 14c.); hence probably modern slang shaft (v.) and the related noun meaning "act of unfair treatment" (1959), though some early sources insist this is from the notion of a wound.

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