Etymology
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idea (n.)

late 14c., "archetype, concept of a thing in the mind of God," from Latin idea "Platonic idea, archetype," a word in philosophy, the word (Cicero writes it in Greek) and the idea taken from Greek idea "form; the look of a thing; a kind, sort, nature; mode, fashion," in logic, "a class, kind, sort, species," from idein "to see," from PIE *wid-es-ya-, suffixed form of root *weid- "to see."

In Platonic philosophy, "an archetype, or pure immaterial pattern, of which the individual objects in any one natural class are but the imperfect copies, and by participation in which they have their being" [Century Dictionary].

Meaning "mental image or picture" is from 1610s (the Greek word for it was ennoia, originally "act of thinking"), as is the sense "concept of something to be done; concept of what ought to be, differing from what is observed." Sense of "result of thinking" first recorded 1640s.

Idée fixe (1836) is from French, literally "fixed idea." Through Latin the word passed into Dutch, German, Danish as idee, which also is found in English dialects. The philosophical sense has been somewhat further elaborated since 17c. by Descartes, Locke, Berkeley, Hume, Kant. Colloquial big idea (as in what's the ...) is from 1908.

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ideogram (n.)
"ideograph," 1837, from ideo-, here as a combining form of idea, + -gram.
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ideate (v.)
c. 1600, "imagine, form ideas," from idea + -ate (2). From 1862 as "to think." Related: Ideated; ideating.
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ideal (adj.)

early 15c., "pertaining to an archetype or model," from Late Latin idealis "existing in idea," from Latin idea in the Platonic sense (see idea). Senses "conceived as perfect; existing only in idea," are from 1610s.

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ideologue (n.)
1815, in reference to the French Revolutionaries, from French ideologue, from Greek idea (see idea) + -logos (see -logy). Earlier form was ideologist (1798).
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ideagenous (adj.)

"generating or giving rise to ideas," 1839; see idea + -genous. A word from early psychology, apparently coined by Dr. Thomas Laycock, house surgeon to York County Hospital [Edinburgh Medical and Surgical Journal, vol. lii]. 

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ideograph (n.)
"character or symbol which suggests an object without expressing its name," 1841, from ideo-, here as a combining form of idea, + -graph "instrument for recording; something written." Related: Ideographic (1822); ideographical.
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ideo- 
word-forming element variously used with reference to images or to ideas, from Greek idea "form; the look of a thing; a kind, sort, nature; mode, fashion," in logic, "a class, kind, sort, species" (see idea).
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ideation (n.)

"process or act of forming ideas," 1829; see idea + -ation. Related: Ideational.

As we say Sensation, we might say also, Ideation; it would be a very useful word; and there is no objection to it, except the pedantic habit of decrying a new term. [James Mill, "Analysis of the Phenomena of the Human Mind," London, 1829]

Related: Ideational.

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-ae 
occasional plural suffix of words ending in -a (see a- (1)), most of which, in English, are from Latin nominative fem. singular nouns (or Greek ones brought up through Latin), which in Latin form their plurals in -ae. But plurals in native -s were established early in English for many of them (such as idea, arena) and many have crossed over since. Now it is not possible to insist on purity one way or the other without breeding monsters.
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