Etymology
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Proto-Indo-European root meaning "to, near, at."

It forms all or part of: abate; ado; ad-; ad hoc; ad lib; adage; adagio; add; adjective; adore; adorn; adult; adverb; advertise; agree; aid; alloy; ally; amontillado; amount; assure; at; atone; exaggerate; paramount; rapport; twit.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit adhi "near;" Latin ad "to, toward;" Old English æt.
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*ag- 

Proto-Indo-European root meaning "to drive, draw out or forth, move."

It forms all or part of: act; action; active; actor; actual; actuary; actuate; agency; agenda; agent; agile; agitation; agony; ambagious; ambassador; ambiguous; anagogical; antagonize; apagoge; assay; Auriga; auto-da-fe; axiom; cache; castigate; coagulate; cogent; cogitation; counteract; demagogue; embassy; epact; essay; exact; exacta; examine; exigency; exiguous; fumigation; glucagon; hypnagogic; interact; intransigent; isagoge; litigate; litigation; mitigate; mystagogue; navigate; objurgate; pedagogue; plutogogue; prodigal; protagonist; purge; react; redact; retroactive; squat; strategy; synagogue; transact; transaction; variegate.

It is the hypothetical source of/evidence for its existence is provided by: Greek agein "to lead, guide, drive, carry off," agon "assembly, contest in the games," agōgos "leader," axios "worth, worthy, weighing as much;" Sanskrit ajati "drives," ajirah "moving, active;" Latin actus "a doing; a driving, impulse, a setting in motion; a part in a play;" agere "to set in motion, drive, drive forward," hence "to do, perform," agilis "nimble, quick;" Old Norse aka "to drive;" Middle Irish ag "battle."

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*agh- 
Proto-Indo-European root meaning "a day" (as a unit of time). The initial d- in Germanic is of obscure origin.

It forms all or part of: adays; Bundestag; daily; daisy; dawn; day; holiday; Reichstag; today.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit dah "to burn," Lithuanian dagas "hot season," Old Prussian dagis "summer."
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*agro- 
Proto-Indo-European root meaning "field;" probably a derivative of root *ag- "to drive, draw out or forth, move."

It forms all or part of: acorn; acre; agrarian; agriculture; agriology; agro-; agronomy; onager; peregrinate; peregrination; peregrine; pilgrim; stavesacre.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit ajras "plain, open country," Greek agros "field," Latin ager (genitive agri) "a field," Gothic akrs, Old English æcer "field."
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*aik- 

Proto-Indo-European root meaning "be master of, possess." 

It forms all or part of: fraught; freight; ought (v.); owe; own.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit ise, iste "he owns, rules," isvara- "owner, lord, ruler;" Avestan ise, is "ruler over," isti- "property, power;" Old English agan "to have, own."

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*aim- 

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

It forms all or part of: emulate; emulation; emulous;  image; imaginary; imagination; imaginative; imagine; imago; imitable; imitate; imitative; imitator; inimitable.

It is the hypothetical source of/evidence for its existence is provided by: Latin imago "image," aemulus "emulous," imitari "to copy, portray, imitate;" Hittite himma- "imitation, substitute."

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*aiw- 

also *ayu-, Proto-Indo-European root meaning "vital force, life; long life, eternity."

It forms all or part of: age; aught (n.1) "something; anything;" aye (adv.) "always, ever;" Ayurvedic; coetaneous; coeval; each; eon; eternal; eternity; ever; every; ewigkeit; hygiene; longevity; medieval; nay; never; no; primeval; sempiternal; tarnation; utopia

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit ayu- "life;" Avestan aiiu "age, life(time);" Greek aiōn "age, vital force; a period of existence, a lifetime, a generation; a long space of time," in plural, "eternity;" Latin aevum "space of time, eternity;" Gothic aiws "age, eternity," Old Norse ævi "lifetime," German ewig "everlasting," Old English a "ever, always."  

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*ak- 

Proto-Indo-European root meaning "be sharp, rise (out) to a point, pierce."

It forms all or part of: acacia; acanthus; accipiter; acer; acerbic; acerbity; acervate; acervulus; acescent; acetic; acid; acicular; acme; acne; acrid; acridity; acrimony; acro-; acrobat; acromegaly; acronym; acrophobia; acropolis; acrostic; acrylic; acuity; aculeate; acumen; acupressure; acupuncture; acute; aglet; ague; Akron; anoxic; awn; coelacanth; dioxin; deoxy-; eager; ear (n.2) "grain part of corn;" edge (n.); egg (v.) "to goad on, incite;" eglantine; epoxy; ester; exacerbation; hammer; hypoxia; mediocre; oxalic; oxide; oxy-; oxygen; oxymoron; paragon; pyracanth; paroxysm; selvage; vinegar.

It is the hypothetical source of/evidence for its existence is provided by: Greek akros "at the end, at the top, outermost; consummate, excellent," akis "sharp point," akē "point of an iron tool," akantha "thorn," akmē "summit, edge," oxys "sharp, bitter;" Sanskrit acri- "corner, edge," acani- "point of an arrow," asrih "edge;" Oscan akrid (ablative singular) "sharply;" Latin acer (fem. acris) "sharp to the senses, pungent, bitter, eager, fierce," acutus "sharp, pointed," acuere "to sharpen," acerbus "harsh, bitter," acere "be sharp, be bitter," acus "a needle, pin," ocris "jagged mountain;" Lithuanian ašmuo "sharpness," akstis "sharp stick;" Old Lithuanian aštras, Lithuanian aštrus "sharp;" Old Church Slavonic ostru, Russian óstryj "sharp;" Old Irish er "high;" Welsh ochr "edge, corner, border;" Old Norse eggja "goad;" Old English ecg "sword;" German Eck "corner."

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*akwa- 

*akwā-, Proto-Indo-European root meaning "water."

It forms all or part of: aqua; aqua-; aqua vitae; aqualung; aquamarine; aquanaut; aquarelle; aquarium; Aquarius; aquatic; aquatint; aqueduct; aqueous; aquifer; Aquitaine; eau; Evian; ewer; gouache; island; sewer (n.1) "conduit."

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit ap "water;" Hittite akwanzi "they drink;" Latin aqua "water, the sea, rain;" Lithuanian upė "a river;" Old English ea "river," Gothic ahua "river, waters." But Boutkan (2005) writes that only the Germanic and Latin words are sure, Old Irish ab is perhaps related, and "the rest of the evidence in Pokorny (1959) is uncertain."

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*al- (1)

Proto-Indo-European root meaning "beyond."

It forms all or part of: adulteration; adultery; alias; alibi; alien; alienate; alienation; allegory; allele;  allergy; allo-; allopathy; allotropy; Alsace; alter; altercation; alternate; alternative; altruism; eldritch; else; hidalgo; inter alia; other; outrage; outrageous; outre; parallax; parallel; subaltern; synallagmatic; ulterior; ultimate; ultra-.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit anya "other, different," arana- "foreign;" Avestan anya-, Armenian ail "another;" Greek allos "other, different, strange;" Latin alius "another, other, different," alter "the other (of two)," ultra "beyond, on the other side;" Gothic aljis "other," Old English elles "otherwise, else," German ander "other."

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