Etymology
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amoeba (n.)
type of microscopic protozoa, 1855, from Modern Latin Amoeba, genus name (1841 in English, said to have been used 1830 by German naturalist Christian Ehrenberg), from Greek amoibe "change, alteration; exchange," which is related to ameibein "to change, exchange," from PIE *(e)meigw-, which is an extended form of root *mei- (1) "to change, go, move" or perhaps a separate root. So called for its constantly changing shape. An earlier popular name for it was proteus (1660s). Related: Amoebic; amoebiform; amoeboid.
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amoebae 
classically correct plural form of amoeba; see -ae.
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amoebaean (adj.)
also amoebean, "alternating, answering alternately," 1650s, from Greek amoibe "change, alteration; exchange" (see amoeba) + -an.
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*mei- (1)
Proto-Indo-European root meaning "to change, go, move," "with derivatives referring to the exchange of goods and services within a society as regulated by custom or law" [Watkins].

It forms all or part of: amiss; amoeba; azimuth; common; commune; communicate; communication; communism; commute; congee; demean; emigrate; emigration; excommunicate; excommunication; immune; immutable; incommunicado; mad; mean (adj.1) "low-quality;" mew (n.2) "cage;" mews; migrate; migration; mis- (1) "bad, wrong;" mistake; Mithras; molt; Mstislav; municipal; munificent; mutable; mutant; mutate; mutation; mutatis mutandis; mutual; permeable; permeate; permutation; permute; remunerate; remuneration; transmutation; transmute; zenith.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit methati "changes, alternates, joins, meets;" Avestan mitho "perverted, false;" Hittite mutai- "be changed into;" Latin mutare "to change," meare "to go, pass," migrare "to move from one place to another," mutuus "done in exchange;" Old Church Slavonic mite "alternately;" Czech mijim "to go by, pass by," Polish mijać "avoid;" Gothic maidjan "to change."
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-a (1)
word-forming element which in English is characteristic of fem. nouns and adjectives of Latin or Greek origin (such as idea, coma, mania, basilica, arena, formula, nebula). From Latin -a (plural -ae) and Greek -a, (plural -ai, Latinized as -ae). The Latin suffix also became Italian -a (plural -e), Spanish -a (plural -as). It is represented in Old English by -u, -e, but even then the suffix was fading and by the time of modern English was totally lost or swallowed into silent final -e-.

It also appears in Romanic words from Latin that have been borrowed into English, such as opera, plaza, armada. It figures in scientific names coined in Modern Latin (amoeba, soda, magnolia, etc.) and is common in geographical names formed according to Latin or Greek models (Asia, Africa, America, Arabia, Florida, etc.)

In English it marks sex only in personal names (Julia, Maria, Alberta) and in a few words from Italian or Spanish where a corresponding male form also is in use (donna, senora).
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oe 

a digraph written also as a ligature (œ) found in Latin words and Greek borrowings into Latin, representing Greek -oi-. Words with -oe- that came early into English from Old French or Medieval Latin usually already had been leveled to -e- (economic, penal, cemetery), but later borrowings directly from Latin or Greek tended to retain it at first (oestrus, diarrhoea, amoeba) as did proper names (Oedipus, Phoebe, Phoenix) and purely technical terms. British English tends to be more conservative with it than American, which has done away with it in all but a few instances.

It also occurred in some native Latin words (foedus "treaty, league," foetere "to stink," hence occasionally in English foetid, foederal, which last was the form in the original publications of the "Federalist" papers). In these it represents an ancient -oi- in Old Latin (for example Old Latin oino, Classical Latin unus), which apparently passed through an -oe- form before being leveled out but was preserved into Classical Latin in certain words, especially those belonging to the realms of law (such as foedus) and religion. These language demesnes, along with the vocabulary of sailors, are the most conservative branches of any language in any time, through a need for precision and immediate comprehension, demonstration of learning, or superstitious dread. But in foetus it was an unetymological spelling in Latin that was picked up in English and formed the predominant spelling of fetus into the early 20c.

The digraph in English also can represent a modified vowel, a mutation or umlaut of -o- in German or Scandinavian words (such as Goethe) and a similar vowel in French words (e.g. oeil "eye," from Latin oculus).

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