Etymology
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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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lot (n.)

Old English hlot "object used to determine someone's share" (anything from dice to straw, but often a chip of wood with a name inscribed on it), also "what falls to a person by lot," from Proto-Germanic *khlutom (source also of Old Norse hlutr "lot, share," Old Frisian hlot "lot," Old Saxon hlot, Middle Dutch, Dutch lot, Old High German hluz "share of land," German Los), from a strong verb (the source of Old English hleotan "to cast lots, obtain by lot; to foretell"). The whole group is of unknown origin.

The object was placed with others in a receptacle (such as a hat or helmet), which was shaken, the winner being the one whose name or mark was on the lot that fell out first. Hence the expression cast lots; to cast (one's) lot with another (1530s, originally biblical) is to agree to share winnings. In some cases the lots were drawn by hand, hence to draw lots. The word was adopted from Germanic into the Romanic languages (Spanish lote, and compare lottery, lotto).

Meaning "choice resulting from the casting of lots" first attested c. 1200. Meaning "share or portion of life" in any way, "that which is given by fate, God or destiny" is from c. 1300. Meaning "number of persons or things of the same kind" is from 1570s (compare Latin mala merx, of persons, literally "a bad lot"). Sense of "plot of land" is first recorded 1630s, American English (distribution of the most desirable properties in new settlements often was determined by casting lots), then especially "parcel of land set aside for a specified purpose" (the Hollywood sense is from 1928). The common U.S. city lot was a rectangle 25 feet wide (along the street) by 100 deep; it was so universal as to be sometimes a unit of measure.

Meaning "group, collection" is 1725, from the notion of auction lots. Lots in the generalized sense of "great many" is attested by 1812; lotsa, colloquial for "lots of," is from 1927; lotta for "lot of" is by 1906.

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foot (n.)

"terminal part of the leg of a vertebrate animal," Old English fot "foot," from Proto-Germanic *fōts (source also of Old Frisian fot, Old Saxon fot, Old Norse fotr, Danish fod, Swedish fot, Dutch voet, Old High German fuoz, German Fuß, Gothic fotus "foot"), from PIE root *ped- "foot." Plural form feet is an instance of i-mutation.

The linear measure was in Old English (the exact length has varied over time), this being considered the length of a man's foot; a unit of measure used widely and anciently. In this sense the plural is often foot. The current inch and foot are implied from measurements in 12c. English churches (Flinders Petrie, "Inductive Metrology"), but the most usual length of a "foot" in medieval England was the foot of 13.2 inches common throughout the ancient Mediterranean. The Anglo-Saxon foot apparently was between the two. All three correspond to units used by the Romans, and possibly all three lengths were picked up by the Anglo-Saxons from the Romano-Britons. "That the Saxon units should descend to mediæval times is most probable, as the Normans were a ruling, and not a working, class." [Flinders Petrie, 1877]. The medieval Paul's Foot (late 14c.) was a measuring standard cut into the base of a column at the old St. Paul's cathedral in London. The metrical foot (late Old English, translating Latin pes, Greek pous in the same sense) is commonly taken to represent one rise and one fall of a foot: keeping time according to some, dancing according to others.

In Middle English also "a person" (c. 1200), hence non-foot "nobody." Meaning "bottom or lowest part of anything eminent or upright" is from c. 1200. Of a bed, grave, etc., from c. 1300. On foot "by walking" is from c. 1300. To get off on the wrong foot is from 1905 (the right foot is by 1907); to put one's best foot foremost first recorded 1849 (Shakespeare has the better foot before, 1596); Middle English had evil-foot (adv.) "through mischance, unluckily." To put one's foot in (one's) mouth "say something stupid" is attested by 1942; the expression put (one's) foot in something "make a mess of it" is from 1823. To have one foot in the grave "be near death" is from 1844. Colloquial exclamation my foot! expressing "contemptuous contradiction" [OED] is attested by 1923, probably euphemistic for my ass in the same sense, which dates to 1796 (also see eyewash).

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