Etymology
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in- (1)
Origin and meaning of in-
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."

In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
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dis- 
Origin and meaning of dis-

word-forming element of Latin origin meaning 1. "lack of, not" (as in dishonest); 2. "opposite of, do the opposite of" (as in disallow); 3. "apart, away" (as in discard), from Old French des- or directly from Latin dis- "apart, asunder, in a different direction, between," figuratively "not, un-," also "exceedingly, utterly." Assimilated as dif- before -f- and to di- before most voiced consonants.

The Latin prefix is from PIE *dis- "apart, asunder" (source also of Old English te-, Old Saxon ti-, Old High German ze-, German zer-). The PIE root is a secondary form of *dwis- and thus is related to Latin bis "twice" (originally *dvis) and to duo, on notion of "two ways, in twain" (hence "apart, asunder").

In classical Latin, dis- paralleled de- and had much the same meaning, but in Late Latin dis- came to be the favored form and this passed into Old French as des-, the form used for compound words formed in Old French, where it increasingly had a privative sense ("not"). In English, many of these words eventually were altered back to dis-, while in French many have been altered back to de-. The usual confusion prevails.

As a living prefix in English, it reverses or negatives what it is affixed to. Sometimes, as in Italian, it is reduced to s- (as in spend, splay, sport, sdain for disdain, and the surnames Spencer and Spence).

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hetero- 
before vowels heter-, word-forming element meaning "other, different," from Greek heteros "the other (of two), another, different; second; other than usual." It is a compound; the first element means "one, at one, together," from PIE root *sem- (1) "one; as one, together with;" the second is cognate with the second element in Latin al-ter, Gothic an-þar, Old English o-ðer "other."

Compounds in classical Greek show the range of the word there: Heterokretes "true Cretan," (that is, of the old stock); heteroglossos "of foreign language;" heterozelos "zealous for one side;" heterotropos "of a different sort or fashion," literally "turning the other way;" heterophron "raving," literally "of other mind."
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en- (1)
word-forming element meaning "in; into," from French and Old French en-, from Latin in- "in, into" (from PIE root *en "in"). Typically assimilated before -p-, -b-, -m-, -l-, and -r-. Latin in- became en- in French, Spanish, Portuguese, but remained in- in Italian.

Also used with native and imported elements to form verbs from nouns and adjectives, with a sense "put in or on" (encircle), also "cause to be, make into" (endear), and used as an intensive (enclose). Spelling variants in French that were brought over into Middle English account for parallels such as ensure/insure, and most en- words in English had at one time or another a variant in in-, and vice versa.
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be- 

word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause, seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by;" see by (prep.)). The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).

The Old English prefix also was used to make transitive verbs and as a privative prefix (as in behead). The sense "on all sides, all about" naturally grew to include intensive uses (as in bespatter "spatter about," therefore "spatter very much," besprinkle, etc.). Be- also can be causative, or have just about any sense required. The prefix was productive 16c.-17c. in forming useful words, many of which have not survived, such as bethwack "to thrash soundly" (1550s) and betongue "to assail in speech, to scold" (1630s).

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sch 

consonant cluster that can represent five distinct sounds in English; it first was used by Middle English writers to render Old English sc-, a sound now generally pronounced (and spelled) "-sh-." Sometimes it was miswritten for ch. It also was taken in from German (schnapps) and Yiddish (schlemiel). In words derived from classical languages (school (n.1)), it represents Latin sch-, Greek skh-, but in some of these words the spelling is a restoration and the pronunciation does not follow it (as in schism; Middle English sisme, cisme).

The Yiddish words with it, often derisive or dismissive, tended to come into 20c. American English. In addition to those with entries here, Saul Bellow used schmegeggy, "Portnoy's Complaint" has schmatte "a ragged garment;" schmeck "a sniff" figures in heroin jargon, and schmutz "dirt, filth" has been used. Directly to English from German also are some specialized words: Schmelz "enamel," schmerz "grief, pain, sorrow," 

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semi- 

word-forming element of Latin origin meaning "half," also loosely, "part, partly; partial, almost; imperfect; twice," from Latin semi- "half" (before vowels often sem-, sometimes further reduced to se- before m-), from PIE *semi- "half" (source also of Sanskrit sami "half," Greek hēmi- "half," Old English sam-, Gothic sami- "half").

The Old English cognate, sam-, was used in such compounds as samhal "in poor health, weakly," literally "half-whole;" samsoden "half-cooked" ('half-sodden'), figuratively "stupid" (compare half-baked); samcucu "half-dead," etymologically "half-alive" (see quick (adj.)); and the lingering survivor, sandblind "dim-sighted" (q.v.).

The Latin element was common in formations from Late Latin, as in semi-gravis "half-drunk," semi-hora "half hour," semi-mortuus "half-dead," semi-nudus "half-naked," semi-vir "half-man, hermaphrodite."

The Latin-derived form in English has been active in forming native words since 15c. Semi-bousi "half-drunk" ('semi-boozy'), now obsolete, was among the earliest (c. 1400). As a noun, semi has variously been short for semi-detached house (by 1912), semi-trailer (by 1942), semi-final (by 1942).

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meta- 
Origin and meaning of meta-

word-forming element of Greek origin meaning 1. "after, behind; among, between," 2. "changed, altered," 3. "higher, beyond;" from Greek meta (prep.) "in the midst of; in common with; by means of; between; in pursuit or quest of; after, next after, behind," in compounds most often meaning "change" of place, condition, etc. This is from PIE *me- "in the middle" (source also of German mit, Gothic miþ, Old English mið "with, together with, among").

The notion of "changing places with" probably led to the senses of "change of place, order, or nature," which was a principal meaning of the Greek word when used as a prefix (but it also denoted "community, participation; in common with; pursuing").

The third, modern, sense, "higher than, transcending, overarching, dealing with the most fundamental matters of," is due to misinterpretation of metaphysics (q.v.) as "science of that which transcends the physical." This has led to a prodigious erroneous extension in modern usage, with meta- affixed to the names of other sciences and disciplines, especially in the academic jargon of literary criticism: Metalanguage (1936) "a language which supplies terms for the analysis of an 'object' language;" metalinguistics (by 1949); metahistory (1957), metacommunication, etc.

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a- (1)

prefix or inseparable particle, a relic of various Germanic and Latin elements.

In words derived from Old English, it commonly represents Old English an "on, in, into" (see on (prep.)), as in alive, above, asleep, aback, abroad, afoot, ashore, ahead, abed, aside, obsolete arank "in rank and file," etc., forming adjectives and adverbs from nouns, with the notion "in, at; engaged in." In this use it is identical to a (2).

It also can represent Middle English of (prep.) "off, from," as in anew, afresh, akin, abreast. Or it can be a reduced form of the Old English past participle prefix ge-, as in aware.

Or it can be the Old English intensive a-, originally ar- (cognate with German er- and probably implying originally "motion away from"), as in abide, arise, awake, ashamed, marking a verb as momentary, a single event. Such words sometimes were refashioned in early modern English as though the prefix were Latin (accursed, allay, affright are examples).

In words from Romanic languages, often it represents reduced forms of Latin ad "to, toward; for" (see ad-), or ab "from, away, off" (see ab-); both of which by about 7c. had been reduced to a in the ancestor of Old French. In a few cases it represents Latin ex.

[I]t naturally happened that all these a- prefixes were at length confusedly lumped together in idea, and the resultant a- looked upon as vaguely intensive, rhetorical, euphonic, or even archaic, and wholly otiose. [OED]
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in- (2)
Origin and meaning of in-

element meaning "into, in, on, upon" (also im-, il-, ir- by assimilation of -n- with following consonant), from Latin in- "in," from PIE root *en "in."

In Old French (and hence in Middle English) this often became en-, which in English had a strong tendency to revert to Latin in-, but not always, which accounts for pairs such as enquire/inquire. There was a native form, which in West Saxon usually appeared as on- (as in Old English onliehtan "to enlighten"), and some of those verbs survived into Middle English (such as inwrite "to inscribe"), but all now seem to be extinct.

Not related to in- (1) "not," which also was a common prefix in Latin, causing confusion: to the Romans impressus could mean "pressed" or "unpressed;" inaudire meant "to hear," but inauditus meant "unheard of;" in Late Latin investigabilis could mean "that may be searched into" or "that cannot be searched into." Latin invocatus was "uncalled, uninvited," but invocare was "to call, appeal to."

The trouble has continued in English; the hesitation over what is meant by inflammable being a commonly cited example. Implume (1610s) meant "to feather," but implumed (c. 1600) meant "unfeathered." Impliable can mean "capable of being implied" (1865) or "inflexible" (1734). Impartible in 17c. could mean "incapable of being divided" or "capable of being imparted." Impassionate can be "free from passion" or it can mean "strongly stirred by passion." Inanimate (adj.) is "lifeless," but Donne uses inanimate (v.) to mean "infuse with life or vigor." Irruption is "a breaking in," but irruptible is "unbreakable."

In addition to improve "use to one's profit," Middle English also had a verb improve meaning "to disprove" (15c.). To inculpate is "to accuse," but inculpable means "not culpable, free from blame." Infestive has meant "troublesome, annoying" (1560s, from infest) and "not festive" (1620s). In Middle English inflexible could mean "incapable of being bent" or "capable of being swayed or moved." In 17c., informed could mean "current in information," formed, animated," or "unformed, formless" ("This was an awkward use" [OED]). Inhabited has meant "dwelt in" (1560s) and "uninhabited" (1610s); inhabitable likewise has been used on opposite senses, a confusion that goes back to Late Latin.

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