Etymology
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beat (v.)

Old English beatan "inflict blows on, strike repeatedly, thrash" (class VII strong verb; past tense beot, past participle beaten), from Proto-Germanic *bautan (source also of Old Norse bauta, Old High German bozan "to beat"), from PIE root *bhau- "to strike."

Past tense beat is from c. 1500, probably not from Old English but a shortening of Middle English beted. Of the heart, c. 1200, from notion of it striking against the breast.

Meaning "to overcome in a contest" is from 1610s (hence the sense of "legally avoid, escape" in beat the charges, etc., attested from c. 1920 in underworld slang). Meaning "be too difficult for" intellectually or physically (by 1870) is behind the shrug-phrase beats me.

Meaning "strike cover to rouse or drive game" (c. 1400) is source of beat around (or about) the bush (1570s), the metaphoric sense of which has shifted from "make preliminary motions" to "avoid, evade." Nautical sense of "make progress against the wind by means of alternate tacks" is from 1670s. Command beat it "go away" first recorded 1906 (though "action of feet upon the ground" was a sense of Old English betan); it is attested in 1903 as newsboy slang for "travel without paying by riding on the outside of a train."

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swear (v.)

Old English swerian "take an oath" (class VI strong verb; past tense swor, past participle sworen), from Proto-Germanic *swērjanan (source also of Old Saxon swerian, Old Frisian swera, Old Norse sverja, Danish sverge, Middle Dutch swaren, Old High German swerien, German schwören, Gothic swaren "to swear"), of uncertain origin, perhaps from a PIE *swer- "to speak, talk, say" (source also of Old Church Slavonic svara "quarrel," Oscan sverrunei "to the speaker").

Also related to the second element in answer. The secondary sense of "use bad language" (early 15c.) probably developed from the notion of "invoke sacred names." Swear off "desist as with a vow" is from 1898. Swear in "install (someone) in office by administration of an oath" is attested from 1700 in modern use, echoing Old English.

[Swearing and cursing] are entirely different things : the first is invoking the witness of a Spirit to an assertion you wish to make ; the second is invoking the assistance of a Spirit, in a mischief you wish to inflict. When ill-educated and ill-tempered people clamorously confuse the two invocations, they are not, in reality, either cursing or swearing ; but merely vomiting empty words indecently. True swearing and cursing must always be distinct and solemn .... [Ruskin, "Fors Clavigera"]
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