Etymology
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at- 
assimilated form of ad- "to, toward, before" before stems beginning in -t-; see ad-. In Old French and Middle English regularly reduced to a-, later restored.
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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 (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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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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at (prep.)

Old English æt, from Proto-Germanic *at (source also of Old Norse, Gothic at, Old Frisian et, Old High German az), from PIE root *ad- "to, near, at." Lost in German and Dutch, which use their equivalent of to; in Scandinavian, however, to has been lost and at fills its place.

At is used to denote relations of so many kinds, and some of these so remote from its primary local sense, that a classification of its uses is very difficult. [OED]

In choosing between at church, in church, etc. at is properly distinguished from in or on by involving some practical connection; a worshipper is at church; a tourist is in the church. In 19c. it was used for points of the compass as regions of a country (at the South) where later tendency was to use in.

The colloquial use of at after where (as in where it's at) is noted in Bartlett (1859). At last is recorded from late 13c.; adverbial phrase at least was in use by 1775. At in Middle English was used freely with prepositions (as in at after, which is in Shakespeare), but this has faded with the exception of at about.

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long (adj.)

Old English lang "having a great linear extent, that extends considerably from end to end; tall; lasting," from Proto-Germanic *langa- (source also of Old Frisian and Old Saxon lang, Old High German and German lang, Old Norse langr, Middle Dutch lanc, Dutch lang, Gothic laggs "long").

The Germanic words perhaps are from PIE *dlonghos- (source also of Latin longus "long, extended; further; of long duration; distant, remote," Old Persian darga-, Persian dirang, Sanskrit dirghah "long"), from root *del- (1) "long" (source also of Greek dolikhos "long," endelekhes "perpetual"). Latin longus (source of prolong, elongate, longitude, etc.) thus is probably cognate with, but not the source of, the Germanic words. The word illustrates the Old English tendency for short "a" to become short "o" before -n- (also retained in bond/band and West Midlands dialectal lond from land and hond from hand).

Also in Old English in reference to time, "drawn out in duration," with overtones of "serious." The old sense of "tall" now appears to be dialectal only, or obsolete. For long "during a long time" is from c. 1300. To be long on something, "have a lot" of it, is from 1900, American English slang. A long vowel (c. 1000) originally was pronounced for an extended time. Mathematical long division is from 1808. Sporting long ball is from 1744, originally in cricket. Long jump as a sporting event is attested from 1864. A long face, one drawn downward in expression of sadness or solemnity, is from 1786. Long in the tooth (1841 of persons) is from horses showing age by recession of gums (but not in this sense until 1870). Long knives, name Native Americans gave to white settlers (originally in Virginia/Kentucky) is from 1774, perhaps a reference to their swords. Long time no see, supposedly imitative of American Indian speech, is first recorded 1919 as Chinese English.

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-long 
adverbial suffix indicating direction, from Old Norse -langr, from langr "long" (adj.); see long (adj.). Displaced native -ling.
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long (adv.)
Old English lange, longe "for a length of time, a long time; far, to a great extent in space," from long (adj.). Old English also had langlice (adv.) "for a long time, long, at length." Longly (adv.) is rarely used. No longer "not as formerly" is from c. 1300; to be not long for this world "soon to die" is from 1714.
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