Etymology
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hat (n.)

Old English hæt "hat, head covering" (variously glossing Latin pileus, galerus, mitra, tiara), from Proto-Germanic *hattuz "hood, cowl" (source also of Frisian hat, Old Norse hattr, höttr "a hood or cowl"), of uncertain etymology; it has been compared with Lithuanian kuodas "tuft or crest of a bird" and Latin cassis "helmet" (but this is said to be from Etruscan).

To throw (one's) hat in the ring was originally (1847) to take up a challenge in prize-fighting. To eat one's hat  (1770), expressing what one will do if something he considers a sure thing turns out not to be, is said to have been originally eat Old Rowley's [Charles II's] hat.

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top-hat (n.)
also tophat, 1875, from top (n.1) + hat.
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hat-box (n.)
also hatbox, 1739, from hat (n.) + box (n.1).
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old hat (adj.)

"out of date," 1911, from old + hat. As a noun phrase, however, it had different sense previously. The "Classical Dictionary of the Vulgar Tongue" (1796) defines it as, "a woman's privities, because frequently felt."

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hard hat (n.)
also hardhat, hard-hat, late 14c., "helmet," from hard (adj.) + hat (n.). From 1935 as "derby hat;" meaning "safety helmet" is from 1953; used figuratively for "construction worker" from 1970.
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hatter (n.)
late 14c., from hat + -er (1). Their association with madness dates to at least 1837.
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high hat (n.)
1839, "tall hat;" also used synechdochically for men who wear such hats; figurative meaning "swelled head" is from 1923. Drum set sense is from 1934.
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hat trick (n.)

in the sports sense, 1879, originally in cricket, "taking three wickets on three consecutive deliveries;" extended to other sports c. 1909, especially ice hockey ("In an earlier contest we had handed Army a 6-2 defeat at West Point as Billy Sloane performed hockey's spectacular 'hat trick' by scoring three goals" ["Princeton Alumni Weekly," Feb. 10, 1941]). So called allegedly because it entitled the bowler to receive a hat from his club commemorating the feat (or entitled him to pass the hat for a cash collection), but the term probably has been influenced by the image of a conjurer pulling objects from his hat (an act attested by 1876). The term was used earlier for a different sort of magic trick:

Place a glass of liquor on the table, put a hat over it, and say, "I will engage to drink every drop of that liquor, and yet I'll not touch the hat." You then get under the table; and after giving three knocks, you make a noise with your mouth, as if you were swallowing the liquor. Then, getting from under the table, say "Now, gentlemen, be pleased to look." Some one, eager to see if you have drunk the liquor, will raise the hat; when you instantly take the glass and swallow the contents, saying, "Gentlemen I have fulfilled my promise: you are all witnesses that I did not touch the hat." ["Wit and Wisdom," London, 1860]
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