Etymology
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swish (v.)

1756 (intransitive); 1799 (transitive), probably imitative of the sound made by something brushing against or through something. Related: Swished; swishing.

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brush (v.1)

late 15c., "clean or rub (clothing) with a brush," also (mid-15c.) "beat with a brush," from brush (n.1). The meaning "move or skim over with a slight contact" is from 1640s. Related: Brushed; brushing. To brush off someone or something, "rebuff, dismiss," is from 1941. To brush up is from c. 1600 as "clean by brushing;" the figurative sense of "revive or refresh one's knowledge" of anything is from 1788.

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brush (v.2)

"move briskly" especially past or against something or someone, 1670s, from earlier sense "to hasten, rush" (c. 1400); probably from brush (n.2) on the notion of a horse, etc., passing through dense undergrowth (compare Old French brosser "to dash (through woods or thickets)," and Middle English noun brush "charge, onslaught, encounter," mid-14c.). But brush (n.1) probably has contributed something to it, and OED suggests the English word could be all or partly onomatopoeic. Related: Brushed; brushing.

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