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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.  

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.

quarter (n.2)

"indulgence or mercy shown to a vanquished foe; exemption from being immediately killed upon being defeated in battle or armed contest," 1610s, presumably from quarter (n.1) or its French equivalent quartier in this meaning, but OED writes that "the precise origin of this sense is obscure ...." It suggests quarter in a now-obsolete sense of "relations with, or conduct towards, another" (attested from 1640s), or possibly quarter in the sense of "place of stay or residence" (compare quarters), on the notion of sending the vanquished to an assigned place until his fate is decided. German quartier, Swedish quarter, Danish kvarteer, etc. probably are from French.

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Definitions of quarter from WordNet
1
quarter (n.)
one of four equal parts;
a quarter of a pound
Synonyms: one-fourth / fourth / one-quarter / fourth part / twenty-five percent / quartern
quarter (n.)
a district of a city having some distinguishing character;
quarter (n.)
(football, professional basketball) one of four divisions into which some games are divided;
both teams scored in the first quarter
quarter (n.)
a unit of time equal to 15 minutes or a quarter of an hour;
a quarter after 4 o'clock
it's a quarter til 4
quarter (n.)
one of four periods into which the school year is divided;
the fall quarter ends at Christmas
quarter (n.)
a fourth part of a year; three months;
unemployment fell during the last quarter
quarter (n.)
one of the four major division of the compass;
the wind is coming from that quarter
quarter (n.)
a quarter of a hundredweight (25 pounds);
quarter (n.)
a quarter of a hundredweight (28 pounds);
quarter (n.)
a United States or Canadian coin worth one fourth of a dollar;
he fed four quarters into the slot machine
quarter (n.)
an unspecified person;
he dropped a word in the right quarter
quarter (n.)
the rear part of a ship;
Synonyms: stern / after part / poop / tail
quarter (n.)
piece of leather that comprises the part of a shoe or boot covering the heel and joining the vamp;
quarter (n.)
clemency or mercy shown to a defeated opponent;
he surrendered but asked for quarter
2
quarter (v.)
provide housing for (military personnel);
Synonyms: billet / canton
quarter (v.)
pull (a person) apart with four horses tied to his extremities, so as to execute him;
in the old days, people were drawn and quartered for certain crimes
Synonyms: draw / draw and quarter
quarter (v.)
divide into quarters;
quarter an apple
quarter (v.)
divide by four; divide into quarters;
From wordnet.princeton.edu