Etymology
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Words related to well

will (v.1)
Old English *willan, wyllan "to wish, desire; be willing; be used to; be about to" (past tense wolde), from Proto-Germanic *willjan (source also of Old Saxon willian, Old Norse vilja, Old Frisian willa, Dutch willen, Old High German wellan, German wollen, Gothic wiljan "to will, wish, desire," Gothic waljan "to choose").

The Germanic words are from PIE root *wel- (2) "to wish, will" (source also of Sanskrit vrnoti "chooses, prefers," varyah "to be chosen, eligible, excellent," varanam "choosing;" Avestan verenav- "to wish, will, choose;" Greek elpis "hope;" Latin volo, velle "to wish, will, desire;" Old Church Slavonic voljo, voliti "to will," veljo, veleti "to command;" Lithuanian velyti "to wish, favor," pa-velmi "I will," viliuos "I hope;" Welsh gwell "better").

Compare also Old English wel "well," literally "according to one's wish;" wela "well-being, riches." The use as a future auxiliary was already developing in Old English. The implication of intention or volition distinguishes it from shall, which expresses or implies obligation or necessity. Contracted forms, especially after pronouns, began to appear 16c., as in sheele for "she will." In early use often -ile to preserve pronunciation. The form with an apostrophe ('ll) is from 17c.
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*wel- (3)
Proto-Indo-European root meaning "to turn, revolve," with derivatives referring to curved, enclosing objects.

It forms all or part of: archivolt; circumvolve; convoluted; convolution; devolve; elytra; evolution; evolve; Helicon; helicopter; helix; helminth; lorimer; ileus; involve; revolt; revolution; revolve; valve; vault (v.1) "jump or leap over;" vault (n.1) "arched roof or ceiling;" volte-face; voluble; volume; voluminous; volute; volvox; volvulus; vulva; wale; walk; wallet; wallow; waltz; well (v.) "to spring, rise, gush;" welter; whelk; willow.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit valate "turns round," ulvam "womb, vulva;" Lithuanian valtis "twine, net," vilnis "wave," apvalus "round;" Old Church Slavonic valiti "roll, welter," vlŭna "wave;" Greek eluein "to roll round, wind, enwrap," eilein "twist, turn, squeeze; revolve, rotate," helix "spiral object;" Latin volvere "to turn, twist;" Gothic walwjan "to roll;" Old English wealwian "roll," weoloc "whelk, spiral-shelled mollusk;" Old High German walzan "to roll, waltz;" Old Irish fulumain "rolling;" Welsh olwyn "wheel."
best (adj.)

Old English beste, reduced by assimilation of -t- from earlier Old English betst "of the highest quality or standing, first, in the best manner." This originally was the superlative of bōt "remedy, reparation" (Middle English bote "advantage, help, profit"), a word now surviving in its simple form only in the expression to boot (see boot (n.2)). Its comparative, better, and superlative, best, have been transferred to good (and in some cases well).

Old English bōt is from Proto-Germanic root *bat-, with comparative *batizon and superlative *batistaz. The superlative form is the source also of Old Frisian, Old Saxon, Middle Dutch best, Old High German bezzist, German best, Old Norse beztr, Gothic batists. Also in Old English as an adverb, "in the most excellent manner."

The best-laid schemes o' mice an' men
     Gang aft agley,
An' lea'e us nought but grief an' pain,
     For promis'd joy!
[Burns, from "To a Mouse, on Turning Her Up in Her Nest With the Plough, November, 1785"]

From late Old English as "of greatest advantage, most suitable." Best-seller as short for "best-selling book" is from 1889, apparently originally in the publishing trade; best friend was in Chaucer (late 14c.). Best girl is first attested 1881, American English; best man is 1814, originally Scottish, replacing groomsman.  

farewell (interj.)
expression at parting, late 14c., from Middle English faren wel, verbal phrase attested by c. 1200 (see fare (v.) + well (adv.)); usually said to the departing person, who replied with good-bye. As a noun, "a good-bye, a leave-taking," by early 15c. Expression to a fare-thee-well "to the last degree" is by 1884, American English.
unwell (adj.)
mid-15c., "somewhat ill," from un- (1) "not" + well (adj.). Similar formation in North Frisian unwel, German unwohl.
wallop (v.)
late 14c., "to gallop," possibly from Old North French *waloper (13c., Old French galoper), from Frankish compound *walalaupan "to run well" (compare Old High German wela "well," see well (adv.); and Old Low Franconian loupon "to run, leap," from Proto-Germanic *hlaupan; see leap (v.)). The meaning "to thrash" (1820) and the noun meaning "heavy blow" (1823) may be separate developments, of imitative origin. Related: Walloped; walloping.
welfare (n.)
c. 1300, from Old English wel faran "condition of being or doing well," from wel (see well (adv.)) + faran "get along" (see fare (v.)). Similar formation in Old Norse velferð. Meaning "social concern for the well-being of children, the unemployed, etc." is first attested 1904; meaning "organized effort to provide for maintenance of members of a group" is from 1918. Welfare state is recorded from 1941.
well-acquainted (adj.)
1728, "having good acquaintance with," from well (adv.) + acquainted.
well-adjusted (adj.)
1735, in reference to mechanisms, etc., from well (adv.) + past participle of adjust (v.). In reference to emotional balance, recorded from 1959.
well-balanced (adj.)
1620s, from well (adv.) + past participle of balance (v.).