Etymology
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last (adj.)

c. 1200, "latest, final, following all others," a contraction of Old English latost (adj.) "slowest, latest," superlative of læt (see late); in some uses from late (adv.). Cognate with Old Frisian lest, Dutch laatst, Old High German laggost, German letzt.

Meaning "last in space, furthest, most remote" is from late 14c.; meaning "most unlikely or unsuitable" is from mid-15c. Meaning "most recent, next before the present" (as in last night, last September) is from late 14c.; latest would be more correct, but idiom rules and the last time I saw her might mean the most recent time this hour or the final time forever.

The biblical last days ("belonging to the end") is attested from late 14c. Last hurrah is from the title of Edwin O'Connor's 1956 novel. Last word "final, definitive statement" is from 1650s. A dying person's last words so called by 1740. As an adjective, last-minute attested from 1913. Last-chance (adj.) is from 1962. Expression if it's the last thing I do, expressing strong determination, is attested by 1905.

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last (n.1)

"wooden model of a human foot used by shoemakers," from Old English læste "shoemaker's last," earlier last "track, footprint, footstep, trace," from Proto-Germanic *laisti- (source also of Old Norse leistr "the foot," Middle Dutch, Dutch leest "form, model, last," Old High German leist "track, footprint," German Leisten "last," Gothic laistjan "to follow"), related to Old English læran "to teach," from PIE root *lois- "furrow, track."

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last (v.)

"endure, go on existing," from Old English læstan "to continue, endure," earlier "follow (a leader), accomplish, carry out, perform," from Proto-Germanic *laistjan "to follow a track" (source also of Gothic laistjan "to follow after," Old Frisian lasta "to fulfill, to pay (duties)," German leisten "to perform, achieve, afford"), from PIE root *lois- "furrow, track." It is related to last (n.1), but not to last (adj.). Related: Lasted; lasting.

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last (n.2)

late Old English, "the last or final man, object, time, etc.," from last (adj.). From late 14c. as "most recent person, latest comer." Also in Middle English as a noun, "duration" (early 14c.), from the verb. Phrase at (the) last is from c. 1200; extended form long last is from 1520s. To the last is from c. 1400.

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last (adv.)

c. 1200, "most recently;" early 13c., "finally, after all others" (contrasted to first), contraction of Old English lætest (adv.), superlative of late (see late).

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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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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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last-ditch (adj.)

"on the last line of defense," 1909, from an image attested by 1715, from a quote attributed to William of Orange (1650-1702), who is said to have uttered it defiantly during the French invasion of 1672; if so, originally in a Netherlands context.

We have no space to enter into the detail of the heroic struggle maintained by the young stadtholder and his faithful Dutchmen; how they laid their country under water, and successfully kept the powerful invader at bay. Once the contest seemed utterly hopeless. William was advised to compromise the matter, and yield up Holland as the conquest of Louis XIV. "No," replied he; "I mean to die in the last ditch." A speech alone sufficient to render his memory immortal. [Agnes Strickland, "Lives of the Queens of England," London, 1847]
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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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present-day (adj.)

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

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