Etymology
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down (adj.)

1560s, "directed downward," from down (adv.). Sense of "depressed mentally" is attested from c. 1600. Slang sense of "aware, wide awake" is attested from 1812. Computer crash sense is from 1965. Down-and-out "completely without resources" is from 1889, American English, from situation of a beaten prizefighter.

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die (v.)

mid-12c., dien, deighen, of sentient beings, "to cease to live," possibly from Old Danish døja or Old Norse deyja "to die, pass away," both from Proto-Germanic *dawjan (source also of Old Frisian deja "to kill," Old Saxon doian, Old High German touwen, Gothic diwans "mortal"), from PIE root *dheu- (3) "to pass away, die, become senseless" (source also of Old Irish dith "end, death," Old Church Slavonic daviti, Russian davit' "to choke, suffer").

It has been speculated that Old English had *diegan, from the same source, but it is not in any of the surviving texts and the preferred words were steorfan (see starve), sweltan (see swelter), wesan dead ("become dead"), also forðgan and other euphemisms.

Languages usually don't borrow words from abroad for central life experiences, but "die" words are an exception; they often are hidden or changed euphemistically out of superstitious dread. A Dutch euphemism translates as "to give the pipe to Maarten."

Regularly spelled dege through 15c., and still pronounced "dee" by some in Lancashire and Scotland. Of plants, "become devitalized, wither," late 14c.; in a general sense of "come to an end" from mid-13c. Meaning "be consumed with a great longing or yearning" (as in dying to go) is colloquial, from 1709. Used figuratively (of sounds, etc.) from 1580s; to die away "diminish gradually" is from 1670s. To die down "subside" is by 1834. Related: Died; dies.

To die out "become extinct" is from 1865. To die game "preserve a bold, resolute, and defiant spirit to the end" (especially of one facing the gallows) is from 1793. Phrase never say die "don't give up or in" is by 1822; the earliest contexts are in sailors' jargon.

"Never look so cloudy about it messmate," the latter continued in an unmoved tone—"Cheer up man, the rope is not twisted for your neck yet. Jack's alive; who's for a row? Never say die while there's a shot in the locker. Whup;" [Gerald Griffin, "Card Drawing," 1842]
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die (n.)

"small cube marked on each face with spots numbering from one to six, used in gaming," early 14c. (as a plural, late 14c. as a singular), from Old French de "die, dice," which is of uncertain origin. Common Romanic (cognates: Spanish, Portuguese, Italian dado, Provençal dat, Catalan dau), perhaps from Latin datum "given," past participle of dare "to give" (from PIE root *do- "to give"), which, in addition to "give," had a secondary sense of "to play" (as a chess piece); or else the notion is "what is given" (by chance or Fortune).

The numbers on the opposite sides always add up to seven; otherwise there is no uniformity to their arrangement. Sense of "engraved stamping block or tool used for stamping a softer material" is from 1690s. Perhaps so called because they often were used in pairs (to impress on both sides, as of a coin).

Figurative phrase the die is cast "the decisive stem is taken" is from 1630s, in reference to the throw of the dice. The expression translates Latin alea iacta est (or iacta alea est), famously uttered by Caesar when he crossed the Rubicon.

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down (n.3)

1710, "a downward movement," from down (adv.). Football sense of "an attempt to advance the ball" is by 1882.

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down (adv.)

"in a descending direction, from a higher to a lower place, degree, or condition," late Old English shortened form of Old English ofdune "downwards," originally of dune "off from (the) hill," from dune "from the hill," dative of dun "hill" (see down (n.2)). The "hill" word is general in Germanic, but this sense development is peculiar to English. As a preposition, "in a descending direction upon or along,"  from late 14c.

To be down on "express disapproval of" is by 1851. Down home is from 1828 as "in one's home region," as an adjective phrase meaning "unpretentious" by 1931, American English. Down the hatch as a toast is from 1931. Down to the wire is 1901, from horse-racing.

Down Under "Australia and New Zealand" attested from 1886; Down East "Maine" is from 1825; Down South "in the Southern states of the U.S." is attested by 1834. Down the road "in the future" is by 1964, U.S. colloquial. Down-to-earth "everyday, ordinary, realistic" is by 1932.

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down (v.)

1560s, "cause to go down," from down (adv.). Meaning "swallow hastily" is by 1860; football sense of "bring down (an opposing player) by tackling" is attested by 1887. Figurative sense of "defeat, get the better of" is by 1898. Related: Downed; downing.

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down (n.1)

"first feathers of a baby bird; soft covering of fowls under the feathers, the under-plumage of birds," used for stuffing pillows and feather-beds, mid-14c., from Old Norse dunn, which is of uncertain origin. Extended in Modern English to the soft hair of the human face and fine soft pubescence upon plants and some fruit.

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down (n.2)

"a hill of moderate elevation and more or less rounded outline," Old English dun "height, hill, moor," from Proto-Germanic *dunaz- (source also of Middle Dutch dunen "sandy hill," Dutch duin), "probably a pre-insular loan-word from Celtic" [Cambridge Dictionary of English Place-Names], in other words, borrowed at a very early period, before the Anglo-Saxon migration, perhaps from PIE root *dheue- "to close, finish, come full circle."

The more general meaning "elevated rolling grassland; high, rolling region not covered by forest" is from c. 1400. Specifically of certain natural pastureland districts of south and southeast England (the Downs) by mid-15c.

The non-English Germanic words tend to mean "dune, sand bank" (see dune), while the Celtic cognates tend to mean "hill, citadel" (compare Old Irish dun "hill, hill fort;" Welsh din "fortress, hill fort;" and second element in place names London, Verdun, etc.). German Düne, French dune, Italian, Spanish duna are said to be loan-words from Dutch.

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sine die 
"indefinitely," Latin, literally "without (fixed) day," from sine "without" (see sans) + ablative singular of dies "day" (from PIE root *dyeu- "to shine").
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die-hard (n.)

also diehard, 1844, in reference to the 57th Regiment of Foot in the British Army, from the verbal phrase die hard "suffer, struggle, or resist in dying," 1784; see die (v.) + hard (adv.). As an adjective, attested from 1871. The  brand of automobile battery, spelled DieHard, was introduced by Sears in 1967.

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