Etymology
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run-down (adj.)

1866, of persons, "to have the health or strength reduced," from the verbal phrase; see run (v.) + down (adv.). From 1896 of places, "dilapidated, shabby, seedy;" by 1894 of clocks, etc., "completely unwound." The earliest sense is "oppressed" (1680s). Compare rundown (n.).

The verbal phrase run down as "have the motive power exhausted" (of clocks, etc.) is by 1761; of persons, etc., "become weak or exhausted," by 1828. To run (something or someone) down "disparage, abuse" is by 1660s. 

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back down (v.)
in figurative sense of "withdraw a charge," 1859, American English, from notion of descending a ladder, etc. (the literal sense by 1849); from back (v.) + down (adv.).
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low-down (adj.)
also low down, lowdown, "vulgar, far down the social scale," 1888, from low (adj.) + down (adv.). Earlier it had meant "humble" (1540s). As a noun, 1915, from the adjective, American English. Low-downer was late 19c. American English colloquial for "poor white; rude, mean person."
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lie-down (n.)
period of rest reclining, 1840, from the verbal phrase (attested from c. 1200); see lie (v.2) + down (adv.).
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tumble-down (adj.)
1791, originally "habitually falling down" and used first of horses, from tumble (v.) + down (adv.); in reference to buildings, "in a dilapidated condition," from 1818.
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up-and-down (adj.)
1610s, from adverbial phrase up and down (c. 1200); see up (adv.) + down (adv.).
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go down (v.)
c. 1300, "droop, descend," from go (v.) + down (adv.). Meaning "decline, fail" is from 1590s. Sense of "to happen" is from 1946, American-English slang. Go down on "perform oral sex on" is from 1916.
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hands down (adv.)

to win something hands down (1855) is from horse racing, from a jockey's gesture of letting the reins go loose in an easy victory.

The Two Thousand Guinea Stakes was not the best contested one that it has been our fortune to assist at. ... [T]hey were won by Meteor, with Scott for his rider; who went by the post with his hands down, the easiest of all easy half-lengths. Wiseacre certainly did the best in his power to spoil his position, and Misdeal was at one time a little vexatious. [The Sportsman, report from April 26, 1840]

Ancient Greek had akoniti "without a struggle, easily," from akonitos (adj.), literally "without dust," specifically "without the dust of the arena."

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upside down (adv.)
late 15c., earlier upsadoun (late 14c.), up so down (c. 1300); the so perhaps meaning "as if." As an adjective from 1866.
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hand-me-down (adj.)
1826, from the verbal phrase; see hand (v.). As a noun from 1874.
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