Etymology
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quarrel (n.1)

[angry dispute] mid-14c., querele, "dispute, altercation," also "ground for complaint," from Old French querele "matter, concern, business; dispute, controversy" (Modern French querelle) and directly from Latin querella "complaint, accusation; lamentation," from queri "to complain, lament," from Proto-Italic *kwese-, of uncertain etymology, perhaps, via the notion of "to sigh," from a PIE root *kues- "to hiss" (source also of Sanskrit svasiti "to hiss, snort"), which is not very compelling, but no better etymology has been offered. 

In Middle English also of armed combat. Old English had sacan. Sense of "angry contention between persons" is from 1570s.

A quarrel is a matter of ill feeling and hard words in view of supposed wrong : it stops just short of blows; any use beyond this is now figurative. [Century Dictionary, 1897]
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quarrelsome (adj.)

"apt to quarrel, given to contention," 1590s, from quarrel (n.1) + -some (1). Related: Quarrelsomely; quarrelsomeness. Older quarrellous (c. 1400) is now obsolete.

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quarry (n.1)

[what is hunted] early 14c., quirre "entrails of deer placed on the hide and given to dogs of the chase as a reward," from Anglo-French quirreie, Old French cuiriee "the spoil, quarry" (Modern French curée), altered (by influence of Old French cuir "skin," from Latin corium "hide"), from Old French corée "viscera, entrails," from Vulgar Latin *corata "entrails," from Latin cor "heart" (from PIE root *kerd- "heart").

The original meaning is obsolete. The sense of "beast of the chase when pursued or slain in a hunt" is by 1610s, also "any object of eager pursuit;" earlier "bird targeted by a hawk or other raptor" (late 15c.).

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quarry (v.)

"to dig or take from a quarry," 1774, from quarry (n.2). Related: Quarried; quarrying.

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quarry (n.2)

"open place where rocks are excavated," late 14c., quarrei (mid-13c. as a place name), from Medieval Latin quareia, a dissimilation of quarreria (mid-13c.), literally "place where stones are squared," from Latin quadrare "to make square," related to quadrus "a square," quattuor "four" (from PIE root *kwetwer- "four"). Quarry-faced, of building stones, is "rough-faced, as though taken right from the quarry" (1849).

The word mine is generally applied to the excavations from which metals, metalliferous ores, and coal are taken ; from quarries are taken all the various materials used for building, as marble, freestone, slate, lime, cement, rock, etc. A quarry is usually open to the day : a mine is generally covered, communicating with the surface by one or more shafts. [Century Dictionary]
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quarryman (n.)

"man occupied in quarrying stones," 1610s, from quarry (n.2) + man (n.). Related: Quarrymen (mid-15c.). Quarriour is attested from 1232 as a surname.

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quart (n.)

liquid measure of capacity equal to one-fourth of a gallon, early 14c., from Old French quarte "a fourth part" (13c.), from Latin quarta (pars), fem. of quartus "the fourth, fourth part" (related to quattuor "four," from PIE root *kwetwer- "four").

Compare Latin quartarius "fourth part," also the name of a small liquid measure (the fourth part of a sextarius), which was about the same as an English pint.

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quartan (adj.)

"having to do with the fourth," especially of attacks of an intermittent fever, etc., "occurring every fourth day" (by inclusive reckoning; now we would say every third day), early 14c., (feuer) quartain, from Old French (fievre) quartaine or cartaine and directly from Latin (febris) quartana, "quartan (fever)," fem. of quartanus "of or belonging to the fourth; of or occurring on the fourth day," from quartus "the fourth, fourth part" (related to quattuor "four," from PIE root *kwetwer- "four").

Also as a noun, "an ague or fever which recurs on the fourth (third) day," late 14c. Under inclusive reckoning, both days of consecutive occurrence are counted (if you have it on Wednesday and again on Saturday, the ancients would count that as "every four days"). 

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quarter (v.)

"to cut in quarters, divide into four equal parts," mid-14c., from quarter (n.1). From late 14c. specifically as the word for a form of criminal punishment (Old English had slitcwealm "death by rending"). Related: Quartered; quartering. Middle English also had a verb quartle "to divide into four parts" (late 14c.).

The meaning "furnish with lodgings, shelter, etc. as a temporary means of living" is recorded from 1590s (see quarters), often specifically "to put up soldiers" under orders from authority.

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quarter (n.1)

c. 1300, "one-fourth of anything; one of four equal parts or divisions into which anything is or may be divided;" often in reference to the four parts into which a slaughtered animal is cut, from Old French quartier, cartier (12c.), from Latin quartarius "fourth part," from quartus "the fourth, fourth part" (related to quattuor "four," from PIE root *kwetwer- "four"). One of the earliest dated references in English is to "parts of the body as dismembered during execution" (c. 1300).

Used of the phases of the moon from early 15c. The phrase quarter of an hour is attested from mid-15c. In Middle English quarter also meant "one of the four divisions of a 12-hour night" (late 14c.), and the quarter of the night meant "nine o'clock p.m." (early 14c.). As a period of time in a football game, from 1911. 

From late 14c. as "one of the four quadrants of the heavens;" hence, from the notion of the winds, "a side, a direction" (c. 1400). In heraldry from mid-14c. as "one of the four divisions of a shield or coat of arms."

Meaning "region, locality, area, place" is from c. 1400. Meaning "distinct portion of a town" (identified by the class or race of people who live there) is first attested 1520s. For military sense, see quarters.

The coin (one fourth of a dollar, originally silver) is peculiar to U.S. and dates to 1783. But quarter could mean "a farthing" (one quarter of a penny) in Middle English (late 14c.), and compare quadrant "a farthing" (c. 1600), and classical Latin quadrans, the name of a coin worth a quarter of an as (the basic unit of Roman currency).

Quarter horse, bred strong for racing on quarter-mile tracks, is recorded by 1834. The word's connection with "four" loosened in Middle English and by 15c. expressions such as six-quartered for "six-sided" are found.  

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