Etymology
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day (n.)

Old English dæg "period during which the sun is above the horizon," also "lifetime, definite time of existence," from Proto-Germanic *dages- "day" (source also of Old Saxon, Middle Dutch, Dutch dag, Old Frisian di, dei, Old High German tag, German Tag, Old Norse dagr, Gothic dags), according to Watkins, from PIE root *agh- "a day."  He adds that the Germanic initial d- is "of obscure origin." But Boutkan says it is from PIE root *dhegh- "to burn" (see fever). Not considered to be related to Latin dies (which is from PIE root *dyeu- "to shine").

Meaning originally, in English, "the daylight hours;" it expanded to mean "the 24-hour period" in late Anglo-Saxon times. The day formerly began at sunset, hence Old English Wodnesniht was what we would call "Tuesday night." Names of the weekdays were not regularly capitalized in English until 17c.

From late 12c. as "a time period as distinguished from other time periods." Day-by-day "daily" is from late 14c.; all day "all the time" is from late 14c.  Day off "day away from work" is attested from 1883; day-tripper first recorded 1897. The days in nowadays, etc. is a relic of the Old English and Middle English use of the adverbial genitive.

All in a day's work "something unusual taken as routine" is by 1820. The nostalgic those were the days is attested by 1907. That'll be the day, expressing mild doubt following some boast or claim, is by 1941. To call it a day "stop working" is by 1919; earlier call it a half-day (1838). One of these days "at some day in the near future" is from late 15c. One of those days "a day of misfortune" is by 1936.

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present-day (adj.)

"current, contemporary, now in existence," 1870, from present (adj.) + day.

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day-book (n.)

also daybook, "book for recording events and transactions of the day," 1570s, from day (n.) + book (n.).

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play-day (n.)

"day given to pastime or diversion, a day exempt from work," c. 1600, from play (v.) + day.

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good-day (n.)
early 12c., "a fortunate day," also, generally, "good fortune;" from good (adj.) + day (n.). As a salutation in parting, haue godne day "have good day" is recorded from c. 1200; good day as a greeting is from late 14c.
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day-dream (n.)

also daydream, "a reverie, pleasant and visionary fancy indulged in when awake," 1680s, from day + dream (n.). As a verb, attested from 1820. Related: Day-dreamer; day-dreaming. Daymare "feeling resembling a nightmare experienced while awake" is from 1737.

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day care (n.)

also daycare, day-care, "care and supervision of young children during the day," especially on behalf of working parents, by 1943, American English, from day + care (n.). Early references are to care for children of women working national defense industry jobs.

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D-day (n.)

1918, "date set for the beginning of a military operation," with D as an abbreviation of day; compare H-hour, also from the same military order of Sept. 7, 1918:

The First Army will attack at H-Hour on D-Day with the object of forcing the evacuation of the St. Mihiel salient. [Field Order No. 8, First Army, A.E.F.]

"They designate the day and hour of the operation when the day and hour have not yet been determined, or where secrecy is essential" [U.S. Army Center of Military History Web site]. Now almost exclusively of June 6, 1944.

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May Day 

"first of May," on which the opening of the season of flowers and fruit formerly was celebrated throughout Europe, mid-13c.; see May + day (n.). May Queen "girl or young woman crowned with flowers and honored as queen at the games held on May Day," seems to be a Victorian re-invented tradition; the phrase Queen of Maij is attested from c. 1500.

May Day's association with communism (and socialism and anarchism) dates to 1890. A U.S. general strike for an eight-hour workday began May 1, 1886, and culminated in the Haymarket bombing affair in Chicago on May 4. By 1890 strikes, protests, and rallies were being held in Europe by socialist and labor organizations on May 1, at first in support of the eight-hour day, more or less in commemoration of the 1886 strike.

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