Etymology
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Latin (adj.)

Old English latin "in Latin," from Latin Latinus "Latin, Roman, in Latin," literally "belonging to Latium," the region of Italy around Rome, a name of uncertain origin. Possibly from PIE root *stela- "to spread, extend," with a sense of "flat country" (as opposed to the mountainous district of the Sabines), or from a prehistoric non-IE language. Old folk etymology connected it with Latin latere "to lie hidden," and a fable of Saturn.

The Latin word also is the source of Spanish and Italian ladino, Dutch latijn, German latein, Irish Gaelic laidionn (n.), Polish lacina, Russian latuinŭ. The more common form in Old English was læden (see Latin (n.)).

In reference to the Roman Catholic Church, 1550s. Used as a designation for "people whose languages descend from Latin" (1856), hence Latin America (1862). The Latin Quarter (French Quartier latin) of Paris, on the south (left) bank of the Seine, was the site of university buildings in the Middle Ages, hence it was the place where Latin was spoken. The surname Latimer means "interpreter," literally "a speaker of Latin."

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Latin (n.)

"the language of the (ancient) Romans," Old English latin "Latin, the language of the Romans; any foreign language," from Latin latinium "the Latin language," noun use of the adjective latinius (see Latin (adj.)). The more common form in Old English was læden, from Vulgar Latin *ladinum, which probably was deformed by influence of Old English leoden "language." For "the Latin language" Old English also had lædenspræc.

In Old French the word was used very broadly, "speech, language:" "What Latin was to the learned, that their tongue was to laymen; hence latino was used for any dialect, even Arabic and the language of birds ...." [Donkin, "Etymological Dictionary of the Romance Languages," 1864].

Roughly speaking, Old Latin is the Latin before the classical period including early authors and inscriptions. Classical Latin flourished from about 75 B.C.E. to about 200 C.E., the Latin of Lucretius, Catullus, Caesar, Cicero, Virgil, Horace, Ovid, Livy, Seneca, etc.; it is the standard Latin of the grammars and dictionaries. Late Latin followed the classical period to about 600 and includes the early church fathers. Medieval Latin was the Latin of the Middle Ages, from about 600 to 1500. Modern Latin is Latin as written from about 1500 on, largely by scientific writers in description and classification. Vulgar Latin was the speech of the Roman home and marketplace, going on concurrently under Classical and Late Latin.

This bit of student doggerel said to have been often scrawled on inside covers of school books, seems to date to 1913 in this or similar wording.

Latin's a dead language —
As dead as it can be;
It killed off all the Romans,
And now it's killing me.
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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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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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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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pig Latin (n.)

childish deformed language (there are many different versions), by 1889 (hog Latin in same sense is attested by 1807).

The animals play quite an important part in the naming [of children's languages], as the hog, dog, fly, goose, pigeon, pig, all give names, with Mr. Hog leading. Among the names the Latins take the lead, and Hog Latin leads the list, being accredited as naming nearly as many languages as all the other names combined. Besides Hog Latin, there is Dog Latin, Pig Latin, Goose Latin, and Bum Latin. Then there is Greekish and Peddlers' French and Pigeon English. ... Very few can give any reason for the naming of the languages. In fact, no one can fully say where the great majority of names came from, for in most cases in the naming the following pretty well expresses the difficulty: "It was born before I was. I can't tell how young I was when I first heard of it." ["The Secret Language of Children," in The North Western Monthly, October 1897]

For the language itself, compare loucherbem, a 20c. French slang similar to pig Latin, which takes its name from the form of the word boucher in that language (which is said to have originated among the Paris butchers).

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quarter day (n.)

mid-15c., "day that begins a quarter of the year," designated as days when rents were paid and contracts and leases began or expired, from quarter (n.1). They were, in England, Lady day (March 25), Midsummer day (June 24), Michaelmas day (Sept. 29), and Christmas day (Dec. 25); in Scotland, keeping closer to the pre-Christian Celtic calendar, they were Candlemas (Feb. 2), Whitsunday (May 15), Lammas (Aug. 1), and Martinmas (Nov. 11). Quarter in the sense "period of three months; one of the four divisions of a year" is recorded from late 14c. Related: Quarter days.

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Anglo-Latin (n.)

Medieval Latin as written in England, 1791, from Anglo- + Latin (n.).

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Latin America 

1862; see Latin (adj.). The notion is the nations whose languages descend from Latin. Related: Latin American (adj.), 1871.

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quarter-deck (n.)

1620s, "the part of the spar-deck of a man-of-war between the poop and the main-mast," originally "a smaller deck above the half-deck," covering about a quarter of the vessel [OED], from quarter (n.1).

"It is used as a promenade by the officers only" [Century Dictionary], hence the colloquial nautical noun quarter-decker (by 1867) "an officer who is more looked upon as a stickler for small points of etiquette than as a thorough seaman."

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