Etymology
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nut (n.)

"the fruit of certain trees and shrubs which have the seed enclosed in a woody covering not opening when ripe," Middle English note, from Old English hnutu, from Proto-Germanic *hnut- (source also of Old Norse hnot, Dutch noot, Old High German hnuz, German Nuss "nut"), from PIE *kneu- "nut" (source also of Latin nux; see nucleus).

Sense of "testicle" is attested by 1915 (nuts). Nut-brown "brown as a ripe, dried nut" is from c. 1300 of animals; c. 1500 of complexions of women. The mechanical nut that goes onto a bolt is first recorded 1610s, from some fancied resemblance (nut was used of other small mechanical pieces since early 15c.). The figurative nuts and bolts "fundamentals" is by 1952. The American English slang sense of "amount of money required for something" is recorded by 1912.  

Meaning "crazy person, crank" is attested from 1903; British form nutter is attested by 1958. Nut-case "crazy person" is from 1959; nut-house "insane asylum" is by 1929. For more on this sense, see nuts. In slang, nut also meant "fashionable or showy young man of affected elegance" [OED], 1904.

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apple (n.)

Old English æppel "apple; any kind of fruit; fruit in general," from Proto-Germanic *ap(a)laz (source also of Old Saxon, Old Frisian, Dutch appel, Old Norse eple, Old High German apful, German Apfel), from PIE *ab(e)l- "apple" (source also of Gaulish avallo "fruit;" Old Irish ubull, Lithuanian obuolys, Old Church Slavonic jabloko "apple"), but the exact relation and original sense of these is uncertain (compare melon).

A roted eppel amang þe holen, makeþ rotie þe yzounde. ["Ayenbite of Inwit," 1340]

In Middle English and as late as 17c., it was a generic term for all fruit other than berries but including nuts (such as Old English fingeræppla "dates," literally "finger-apples;" Middle English appel of paradis "banana," c. 1400). Hence its grafting onto the unnamed "fruit of the forbidden tree" in Genesis. Cucumbers, in one Old English work, are eorþæppla, literally "earth-apples" (compare French pomme de terre "potato," literally "earth-apple;" see also melon). French pomme is from Latin pomum "apple; fruit" (see Pomona).

As far as the forbidden fruit is concerned, again, the Quran does not mention it explicitly, but according to traditional commentaries it was not an apple, as believed by Christians and Jews, but wheat. [Seyyed Hossein Nasr, "The Heart of Islam: Enduring Values for Humanity," 2002]

Apple of Discord (c. 1400) was thrown into the wedding of Thetis and Peleus by Eris (goddess of chaos and discord), who had not been invited, and inscribed kallisti "To the Prettiest One." Paris, elected to choose which goddess should have it, gave it to Aphrodite, offending Hera and Athene, with consequences of the Trojan War, etc.

Apple of one's eye (Old English), symbol of what is most cherished, was the pupil, supposed to be a globular solid body. Apple-polisher "one who curries favor" first attested 1928 in student slang. The image in the phrase upset the apple cart "spoil the undertaking" is attested from 1788. Road-apple "horse dropping" is from 1942.

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apple-tree (n.)
Old English æpeltreow; see apple + tree (n.).
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nut-cracker (n.)

also nutcracker, "instrument used for cracking hard-shelled nuts," 1540s, from nut (n.) + agent noun from crack (v.). Hence also "toy having a grotesque human head, in the mouth of which a nut is placed to be cracked by a screw or lever." The ballet was first performed in 1892, based on Dumas père's rendition of E.T.A. Hoffmann's 1816 story "Nussknacker und Mausekönig."

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pine-nut (n.)

"pine-cone," a sense now obsolete; also "edible seed-kernel of several species of pine," Old English pinhnyte; see pine (n.) + nut (n.).

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apple-pie (n.)
1580s, from apple + pie; noted by 1893 as a typical American dish. Apple-pie bed as a name for a childish prank is recorded from 1781; supposedly from the way of making apple turnovers, but some think it a folk etymology of French nappe pliée "folded sheet."
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Adam's apple (n.)
bulge in the throat caused by the cartilage of the larynx, 1731, corresponding to Latin pomum Adami, perhaps an inexact translation of Hebrew tappuah haadam, literally "man's swelling," from ha-adam "the man" + tappuah "anything swollen." The reference is to the legend that a piece of the forbidden fruit (commonly believed to have been an apple) that Eve gave Adam stuck in his throat. It is more prominent in men than women. The term is mentioned early 15c. as the name of an actual oriental and Mediterranean fruit, a variety of lime with an indentation fancied to resemble the marks of Adam's teeth.
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Big Apple (n.)
"New York City," 1909 (but popularized by 1970s tourism promotion campaign), apparently from jazz musicians' use of apple for any city, especially a Northern one.
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love-apple (n.)

old name for "tomato," 1570s, translating French pomme d'amour, corresponding to German Liebesapfel, etc., but the alleged aphrodisiac qualities that supposedly inspired the name seem far-fetched. The phrase also has been explained as a mangled transliteration of the Italian name pomo d'oro (by 1560s), taken as from adorare "to adore," but probably rather from d'or "of gold" (the earliest tomatoes brought to Italy in the mid-1500s apparently were of the yellow or orange variety), or, less likely, pomo de'Mori or Spanish pome dei Moro, literally "Moorish apple."

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apple-sauce (n.)
also applesauce, by 1739, American English, from apple + sauce (n.). Slang meaning "nonsense" is attested from 1921 and was noted as a vogue word early 1920s. Mencken credits it to cartoonist T.A. ("Tad") Dorgan. DAS suggests the word was thus used because applesauce was cheap fare served in boardinghouses.
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