Etymology
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Words related to band

*bhendh- 
Proto-Indo-European root meaning "to bind."

It forms all or part of: band; bandanna; bend; bind; bindle; bond; bund; bundle; cummerbund; ribbon; woodbine.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit badhnati "binds," bandhah "a tying, bandage;" Old Persian bandaka- "subject;" Lithuanian bendras "partner;" Middle Irish bainna "bracelet;" Old English bendan "to bend a bow, confine with a string," bindan "to bind," Gothic bandi "that which binds."
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bond (n.)

early 13c., "anything that binds, fastens, or confines," phonetic variant of band (n.1) and at first interchangeable with it. For vowel change, see long (adj.); also influenced by unrelated Old English bonda "householder," literally "dweller" (see bond (adj.)).

It preserves more distinctly than band the connection with bind and bound (adj.1) and is now the main or only form in the sense of "restraining or uniting force." From early 14c. as "an agreement or covenant;" from late 14c. as "a binding or uniting power or influence." Legalistic sense "an instrument binding one to pay a sum to another" first recorded 1590s. Meaning "a method of laying bricks in courses" is from 1670s. In chemistry, of atoms, by 1900.

bend (n.2)
"broad diagonal band in a coat-of-arms, etc.," mid-14c., from earlier sense of "thin, flat strap for wrapping round," from Old English bend "fetter, shackle, chain," from PIE *bhendh- "to bind" (see bend (v.)). Probably in part also from Old French bende (Modern French bande) and Medieval Latin benda, both from Germanic. Ordinarily running from the right top to the left bottom; the bend sinister runs along the other diagonal.
arm-band (n.)

"band or bracelet for the arm," by 1782; see from arm (n.1) + band (n.1).

bandage (n.)

"strip of soft cloth or other material used in binding wounds, stopping bleeding, etc.," 1590s, from French bandage (16c.), from Old French bander "to bind," from bande "a strip" (see band (n.1)).

Band-Aid (n.)
trademark name (Johnson & Johnson) for a stick-on gauze pad or strip, by 1922. See band (n.1) + aid (n.). The British equivalent was Elastoplast. Figurative sense of "temporary or makeshift solution to a problem, pallative" (often lower case, sometimes bandaid) is attested by 1968; as an adjective in this sense, by 1970.
bandbox (n.)
"light box of pasteboard or thin wood, originally made to hold the starched bands worn as collars in 17c. men's and women's dress, 1630s, from band (n.1) + box (n.1). Later used for other light articles of attire, but the name stuck. Typical of something fragile and flimsy, but it also was figurative of smallness and of neat, clean condition.
bandeau (n.)
1706, "headband," from French bandeau, from Old French bandel, bendel "bandage, binding" (12c.), diminutive of bande "band" (see band (n.1)). As a style of women's top or bra by 1968.
band-saw (n.)
also bandsaw, "endless band of steel with a serrated edge," 1847, from band (n.1) + saw (n.1). Said to have been invented 1809 by William Newberry of London.
bandwidth (n.)
1930, in electronics, from band (n.1) + width.