Etymology
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deal (n.1)

"a part or portion," Middle English del, from from Old English dæl "a part of a whole, a share;" with qualification (great, etc.), "an extent, degree, quantity, amount," from Proto-Germanic *dailaz (source also of Old Norse deild, Old Frisian del "part; juridical district," Dutch deel, Old High German and German teil, Gothic dails "part, share, portion"), from PIE *dail- "to divide" (source also of Old Church Slavonic delu, Lithuanian dalis "part"), ‌‌perhaps a Northern Indo-European extended form of the root *da- "to divide," or perhaps a substratum word.

Formerly used in many senses now taken by part. Meaning "a share (of something), one's allotted portion" is from c. 1200. Business sense of "transaction, bargain" is 1837, originally slang, from the older sense of "arrangement among a number of persons for mutual advantage." In American history, New Deal is from Franklin D. Roosevelt's speech of July 2, 1932 (the phrase itself is by 1834). Big deal is from 1928 as "important transaction;" ironic use first recorded 1951 in "Catcher in the Rye." Deal-breaker is attested by 1975.

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great (adj.)

Old English great "big, tall, thick, stout, massive; coarse," from West Germanic *grauta- "coarse, thick" (source also of Old Saxon grot, Old Frisian grat, Dutch groot, German groß "great"). If the original sense was "coarse," it is perhaps from PIE root *ghreu- "to rub, grind," via the notion of "coarse grain," then "coarse," then "great;" but "the connextion is not free from difficulty" [OED].

It took over much of the sense of Middle English mickle, and itself now is largely superseded by big and large except in reference to non-material things. In the sense of "excellent, wonderful" great is attested from 1848.

Great White Way "Broadway in New York City" is from 1901, in reference to brilliant street illumination. The Great Lakes of North America so called by 1726, perhaps 1690s. Great Spirit "high deity of the North American Indians," 1703, originally translates Ojibwa kitchi manitou. The Great War originally (1887) referred to the Napoleonic Wars, later (1914) to what we now call World War I (see world).

"The Great War" — as, until the fall of France, the British continued to call the First World War in order to avoid admitting to themselves that they were now again engaged in a war of the same magnitude. [Arnold Toynbee, "Experiences," 1969]

Also formerly with a verb form, Old English greatian "to become enlarged," Middle English greaten "to become larger, increase, grow; become visibly pregnant," which became archaic after 17c.

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deal (n.2)

"plank or board," especially of fir or pine, late 14c., dele, from Low German (compare Middle Low German dele), from Proto-Germanic *theljon." From late 13c. in surnames. An Old English derivative was þelu"hewn wood, board, flooring."

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deal (v.)

Middle English delen, from Old English dælan "to divide, distribute, separate;" hence "to share with others, bestow, dispense," and also "take part in, have to do with," from Proto-Germanic *dailjanan (source also of Old Saxon deljan, Old Frisian dela "to divide, distribute," Middle Dutch, Dutch deelen, German teilen, Gothic dailjan),from PIE *dail- "to divide," ‌‌perhaps a Northern Indo-European extended form of root *da- "to divide," or a word from a substrate language.

Meaning "to deliver (to another) as his share" is from c. 1300. Meaning "to distribute cards before a game" is from 1520s (the associated noun meaning "distribution of cards before a game" is from c. 1600). Hence colloquial deal (someone) in "include in an undertaking" (1942).

To deal with "handle, act toward (in some way)" is attested from mid-15c., from the notion of "engage in mutual intercourse, have to do with;" in late 14c. the phrase also mean "have sexual intercourse with." Related: Dealt; dealing.

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great- 
word-forming element denoting "kinship one degree further removed," early 15c. (in great uncle), from great (adj.), based on similar use of French grand (see grand-). An Old English way of saying "great-grandfather" was þridda fæder, literally "third father;" in early Middle English furþur ealdefader was used (12c.).
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big deal (n.)
from 1860s as "a good deal, a large amount;" by 1878 in financial speculation, originally in California publications; see deal (n.1). As an ironic expression, popular in American English from c. 1965, perhaps a translated Yiddishism (such as a groyser kunst).
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Great Britain 
c. 1400, Grete Britaigne "the land of the Britons before the English conquest" (as opposed to Brittany), also "England and Wales;" see great (adj.) + Britain.
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great-hearted (adj.)
"of noble courage," late 14c., from great (adj.) + -hearted.
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