Etymology
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*kweie- 
*kweiə-, also *kwyeə-, Proto-Indo-European root meaning "to rest, be quiet."

It forms all or part of: acquiesce; acquit; awhile; coy; quiesce; quiescent; quiet; quietism; quietude; quietus; quit; quitclaim; quite; quit-rent; quittance; requiescat; requiem; requite; while; whilom.

It is the hypothetical source of/evidence for its existence is provided by: Avestan shaitish "joy," shaiti- "well-being," shyata- "happy;" Old Persian šiyatish "joy;" Latin quies "rest, repose, quiet;" Old Church Slavonic po-koji "rest;" Old Norse hvild "rest."
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*deu- (2)
Proto-Indo-European root meaning "to do, perform; show favor, revere."

It forms all or part of: beatific; beatify; beatitude; Beatrice; beau; beauty; Bella; belle; beldam; belladonna; belvedere; bene-; benedict; Benedictine; benediction; benefactor; beneficiary; benefice; beneficence; benefit; benevolent; benign; bonanza; bonbon; bonhomie; bonito; bonjour; bonny; bonus; boon (adj.); bounty; debonair; embellish.

It is the hypothetical source of/evidence for its existence is provided by: Latin bene (adv.) "well, in the right way, honorably, properly," bonus "good," bellus "handsome, fine, pretty," and possibly beatus "blessed," beare "to make blessed."
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*(s)mer- (1)

Proto-Indo-European root meaning "to remember." 

It forms all or part of: commemorate; commemoration; mourn; memo; memoir; memorable; memorandum; memorial; memorious; memorize; memory; remember.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit smarati "remembers;" Avestan mimara "mindful;" Greek merimna "care, thought," mermeros "causing anxiety, mischievous, baneful;" Latin memoria "memory, remembrance, faculty of remembering," memor "mindful, remembering;" Serbo-Croatian mariti "to care for;" Welsh marth "sadness, anxiety;" Old Norse Mimir, name of the giant who guards the Well of Wisdom; Old English gemimor "known," murnan "to mourn, remember sorrowfully;" Dutch mijmeren "to ponder."

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*sol- 

also solə-, Proto-Indo-European root meaning "whole, well-kept." 

It forms all or part of: catholic; consolidate; consolidation; holism; holo-; holocaust; Holocene; hologram; holograph; insouciant; safe; safety; sage (n.1) kind of herb; salubrious; salutary; salute; salvage; salvific; salvo "simultaneous discharge of guns;" save (v.) "deliver from danger;" save (prep.) "except;" solder; soldier; solemn; solicit; solicitous; solid; solidarity; solidity; sou.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit sarvah "uninjured, intact, whole;" Avestan haurva- "uninjured, intact;" Old Persian haruva-; Greek holos "whole;" Latin salvus "uninjured, in good health, safe," salus "good health," solidus "solid;" Armenian olj "whole, healthy."  

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*al- (2)
Proto-Indo-European root meaning "to grow, nourish."

It forms all or part of: abolish; adolescent; adult; alderman; aliment; alimony; Alma; alma mater; alt (2) "high tone;" alti-; altimeter; altitude; alto; alumnus; auld; coalesce; elder (adj., n.1); eldest; Eldred; enhance; exalt; haught; haughty; hautboy; hawser; oboe; old; proletarian; proliferation; prolific; world.

It is the hypothetical source of/evidence for its existence is provided by: Greek aldaino "make grow, strengthen," althein, althainein "to get well;" Latin alere "to feed, nourish, suckle; bring up, increase," altus "high," literally "grown tall," almus "nurturing, nourishing," alumnus "fosterling, step-child;" Gothic alþeis, Dutch oud, German alt "old;" Gothic alan "to grow up," Old Norse ala "to nourish;" Old Irish alim "I nourish."
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*wal- 

Proto-Indo-European root meaning "to be strong."

It forms all or part of: ambivalence; Arnold; avail; bivalent; convalesce; countervail; Donald; equivalent; evaluation; Gerald; Harold; invalid (adj.1) "not strong, infirm;" invalid (adj.2) "of no legal force;" Isold; multivalent; polyvalent; prevalent; prevail; Reynold; Ronald; valediction; valence; Valerie; valetudinarian; valiance; valiant; valid; valor; value; Vladimir; Walter; wield.

It is the hypothetical source of/evidence for its existence is provided by: Latin valere "be strong, be well, be worth;" Old Church Slavonic vlasti "to rule over;" Lithuanian valdyti "to have power;" Celtic *walos- "ruler," Old Irish flaith "dominion," Welsh gallu "to be able;" Old English wealdan "to rule," Old High German -walt, -wald "power" (in personal names), Old Norse valdr "ruler."

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*bhreu- 
also *bhreuə-, *bhreəu-, Proto-Indo-European root meaning "to boil, bubble, effervesce, burn," with derivatives referring to cooking and brewing.

It forms all or part of: barm; barmy; bourn (n.1) "small stream;" braise; bratwurst; brawn; brawny; braze (v.1) "to expose to the action of fire;" brazier; Brazil; bread; breed; brew; broth; broil (v.2) "to quarrel, brawl;" brood; effervesce; effervescence; effervescent; embroil; ferment; fervent; fervid; fervor; imbroglio.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit bhurnih "violent, passionate;" Greek phrear "well, spring, cistern;" Latin fervere "to boil, foam," Thracian Greek brytos "fermented liquor made from barley;" Russian bruja "current;" Old Irish bruth "heat;" Old English breowan "to brew," beorma "yeast;" Old High German brato "roast meat."
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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."
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*ghos-ti- 

Proto-Indo-European root meaning "stranger, guest, host," properly "someone with whom one has reciprocal duties of hospitality," representing "a mutual exchange relationship highly important to ancient Indo-European society" [Watkins]. But as strangers are potential enemies as well as guests, the word has a forked path.

The word ghos-ti- was thus the central expression of the guest-host relationship, a mutual exchange relationship highly important to ancient Indo-European society. A guest-friendship was a bond of trust between two people that was accompanied by ritualized gift-giving and created an obligation of mutual hospitality and friendship that, once established, could continue in perpetuity and be renewed years later by the same parties or their descendants. [Calvert Watkins, "American Heritage Dictionary of Indo-European Roots"]

It forms all or part of: Euxine; guest; hospice; hospitable; hospital; hospitality; hospodar; host (n.1) "person who receives guests;" host (n.2) "multitude;" hostage; hostel; hostile; hostility; hostler; hotel; Xenia; xeno-; xenon.

It is the hypothetical source of/evidence for its existence is provided by: Greek xenos "guest, host, stranger;" Latin hostis, in earlier use "a stranger," in classical use "an enemy," hospes "host;" Old Church Slavonic gosti "guest, friend," gospodi "lord, master;" Old English gæst, "chance comer, a stranger."

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*wegh- 

Proto-Indo-European root meaning "to go, move, transport in a vehicle."

The root wegh-, "to convey, especially by wheeled vehicle," is found in virtually every branch of Indo-European, including now Anatolian. The root, as well as other widely represented roots such as aks- and nobh-, attests to the presence of the wheel — and vehicles using it — at the time Proto-Indo-European was spoken. [Watkins, p. 96]

It forms all or part of: always; away; convection; convey; convex; convoy; deviate; devious; envoy; evection; earwig; foy; graywacke; impervious; invective; inveigh; invoice; Norway; obviate; obvious; ochlocracy; ogee; pervious; previous; provection; quadrivium; thalweg; trivia; trivial; trivium; vector; vehemence; vehement; vehicle; vex; via; viaduct; viatic; viaticum; vogue; voyage; wacke; wag; waggish; wagon; wain; wall-eyed; wave (n.); way; wee; weigh; weight; wey; wiggle.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit vahati "carries, conveys," vahitram, vahanam "vessel, ship;" Avestan vazaiti "he leads, draws;" Greek okhos "carriage, chariot;" Latin vehere "to carry, convey," vehiculum "carriage, chariot;" Old Church Slavonic vesti "to carry, convey," vozŭ "carriage, chariot;" Russian povozka "small sled;" Lithuanian vežu, vežti "to carry, convey," važis "a small sled;" Old Irish fecht "campaign, journey," fen "carriage, cart;" Welsh gwain "carriage, cart;" Old English wegan "to carry;" Old Norse vegr, Old High German weg "way;" Middle Dutch wagen "wagon."

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