Etymology
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vigil (n.)
c. 1200, "eve of a religious festival" (an occasion for devotional watching or observance), from Anglo-French and Old French vigile "watch, guard; eve of a holy day" (12c.), from Latin vigilia "a watch, watchfulness," from vigil "watchful, awake, on the watch, alert," from PIE root *weg- "to be strong, be lively." Meaning "watch kept on a festival eve" in English is from late 14c.; general sense of "occasion of keeping awake for some purpose" is recorded from 1711.
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*weg- 
Proto-Indo-European root meaning "to be strong, be lively."

It forms all or part of: awake; bewitch; bivouac; invigilate; reveille; surveillance; vedette; vegetable; velocity; vigil; vigilant; vigilante; vigor; waft; wait; wake (v.) "emerge or arise from sleep;" waken; watch; Wicca; wicked; witch.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit vajah "force, strength," vajayati "drives on;" Latin vigil "watchful, awake," vigere "be lively, thrive," velox "fast, lively," vegere "to enliven," vigor "liveliness, activity;" Old English wacan "to become awake," German wachen "to be awake," Gothic wakan "to watch."
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vigilance (n.)

1560s, from French vigilance (16c.), from Latin vigilantia "wakefulness, watchfulness, attention," from vigil "watchful, awake" (from PIE root *weg- "to be strong, be lively"). Related: Vigilancy (1530s).

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vigilant (adj.)

late 15c., from French vigilant or directly from Latin vigilantem (nominative vigilans) "watchful, anxious, careful," present participle of vigilare "to watch, keep awake, not to sleep, be watchful," from vigil "watchful, awake" (from PIE root *weg- "to be strong, be lively"). Related: Vigilantly.

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vigilante (n.)
"member of a vigilance committee," 1856, American English, from Spanish vigilante, literally "watchman," from Latin vigilantem (nominative vigilans) "watchful, anxious, careful," from vigil "watchful, awake" (from PIE root *weg- "to be strong, be lively"). Vigilant man in same sense is attested from 1824 in a Missouri context. Vigilance committees kept informal rough order on the U.S. frontier or in other places where official authority was imperfect.
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wake (n.2)
"state of wakefulness," Old English -wacu (in nihtwacu "night watch"), related to watch (n.); and partly from Old Norse vaka "vigil, eve before a feast" (which is related to vaka "be awake" and cognate with Old High German wahta "watch, vigil," Middle Dutch wachten "to watch, guard"), from PIE root *weg- "to be strong, be lively." Meaning "a sitting up at night with a corpse" is attested from early 15c. (the verb in this sense is recorded from mid-13c.; as a noun lichwake is from late 14c.). The custom largely survived as an Irish activity. Wakeman (c. 1200), which survives as a surname, was Middle English for "watchman."
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vedette (n.)
"mounted sentinel placed in advance of an outpost," 1680s, from French vedette (16c.), from Italian (Florentine) vedetta "watch tower, peep hole," probably from vedere "to see," from Latin videre "to see" (from PIE root *weid- "to see"), or else from Latin vigil "watchful, awake," from PIE root *weg- "to be strong, be lively."
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surveillance (n.)
1802, from French surveillance "oversight, supervision, a watch," noun of action from surveiller "oversee, watch" (17c.), from sur- "over" (see sur- (1)) + veiller "to watch," from Latin vigilare, from vigil "watchful" (from PIE root *weg- "to be strong, be lively"). Seemingly a word that came to English from the Terror in France ("surveillance committees" were formed in every French municipality in March 1793 by order of the Convention to monitor the actions and movements of suspect persons, outsiders, and dissidents).
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death-watch (n.)

"a vigil beside a dying person," 1865, from death + watch (n.) "a watching." The death-watch beetle (1660s) inhabits houses, makes a ticking noise like a pocket-watch, and was superstitiously supposed to portend death.

FEW ears have escaped the noise of the death-watch, that is, the little clicking sound heard often in many rooms, somewhat resembling that of a watch; and this is conceived to be of an evil omen or prediction of some person's death: wherein notwithstanding there is nothing of rational presage or just cause of terror unto melancholy and meticulous heads. For this noise is made by a little sheathwinged grey insect, found often in wainscot benches and wood-work in the summer. [Browne, "Vulgar Errors"] 
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