Etymology
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rat (n.)

"a rodent of some of the larger species of the genus Mus," late Old English ræt "rat," a word of uncertain origin. Similar words are found in Celtic (Gaelic radan), Romanic (Medieval Latin ratus, Italian ratto, Spanish rata, Old French rat) and Germanic (Old Saxon ratta; Middle Dutch ratte, Dutch rat; German Ratte, dialectal Ratz; Swedish råtta, Danish rotte) languages, but their connection to one another and the ultimate source of the word are unknown. In its range and uncertain origin, it is much like cat.

Perhaps from Vulgar Latin *rattus, but Weekley thinks this is of Germanic origin, "the animal having come from the East with the race-migrations" and the word passing thence to the Romanic languages. American Heritage and Tucker connect Old English ræt to Latin rodere and thus to PIE root *red- "to scrape, scratch, gnaw," source of rodent (q.v.). Klein says there is no such connection and suggests a possible cognate in Greek rhine "file, rasp." Weekley connects the English noun and the Latin verb with a question mark and OED says it is "probable" that the rat word spread from Germanic to Romanic, but takes no position on further etymology. The common Middle English form was ratton, from augmented Old French form raton. Applied to rat-like species on other continents from 1580s.

The distinction between rat and mouse, in the application of the names to animals everywhere parasitic with man, is obvious and familiar. But these are simply larger and smaller species of the same genus, very closely related zoologically, and in the application of the two names to the many other species of the same genus all distinction between them is lost. [Century Dictionary]

Applied since 12c. (in surnames) to persons held to resemble rats or share some characteristic or quality with them. Specific sense of "one who abandons his associates for personal advantage" (1620s) is from the belief that rats leave a ship about to sink or a house about to fall, and this led to the meaning "traitor, informant" (1902).

To smell a rat "to be put on the watch by suspicion as the cat by the scent of a rat; to suspect danger" [Johnson] is from 1540s.  _____-rat, "person who frequents _____" (in earliest reference dock-rat) is from 1864.

RATS. Of these there are the following kinds: a black rat and a grey rat, a py-rat and a cu-rat. ["Dictionary of the Vulgar Tongue," Grose, 1788]  
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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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run (v.)

Old English, "move swiftly by using the legs, go on legs more rapidly than walking," also "make haste, hurry; be active, pursue or follow a course," and, of inanimate things, "to move over a course."

The modern verb is a merger of two related Old English words, in both of which the initial two letters sometimes switched places. The first is intransitive rinnan, irnan "to run, flow, run together" (past tense ran, past participle runnen), which is cognate with Middle Dutch runnen, Old Saxon, Old High German, Gothic rinnan, German rinnen "to flow, run."

The second is Old English transitive weak verb ærnan, earnan "ride, run to, reach, gain by running" (probably a metathesis of *rennan), from Proto-Germanic *rannjanan, causative of the root *ren- "to run." This is cognate with Old Saxon renian, Old High German rennen, German rennen, Gothic rannjan.

Watkins says both are from PIE *ri-ne-a-, nasalized form of root *rei- "to run, flow," but Boutkan's sources find this derivation doubtful based on the poor attestation of supposed related forms, and he lists it as of "No certain IE etymology."

Of streams, etc., "to flow," from late Old English. From c. 1200 as "take flight, retreat hurriedly or secretly." Phrase run for it "take flight" is attested from 1640s.

Also from c. 1200 as "compete in a race." Extended to "strive for any ends," especially "enter a contest for office or honors, stand as a candidate in an election" (1826, American English).

Of any sort of hurried travel, c. 1300. From early 13c. as "have a certain direction or course." By c. 1300 as "keep going, extend through a period of time, remain in existence." Specifically of theater plays by 1808. Of conveyances, stage lines, etc., "perform a regular passage from place to place" by 1817.

Of machinery or mechanical devices, "go through normal or allotted movements or operation," 1560s. Of colors, "to spread in a fabric when exposed to moisture," 1771. Of movie film, "pass between spools," hence "be shown," by 1931.

The meaning "carry on" (a business, etc.) is by 1861, American English; hence extended senses of "look after, manage." As "publish or print in a newspaper or magazine," by 1884. 

Many senses are via the notion of "pass into or out of a certain state." To run dry "cease to yield water or milk" (1630s). In commerce, "be of a specified price, size, etc.," by 1762. To run low "be nearly exhausted" is by 1712; to run short "exhaust one's supply" is from 1752; to run out of in the same sense is from 1713. To run on "keep on, continue without pause or change" is from 1590s.

The transitive sense of "cause to run" was in Old English. By late 15c. as "to pierce, stab," hence 1520s as "thrust through or into something." The meaning "enter (a horse) in a race" is from 1750. The sense of "cause a mechanical device to keep moving or working" is by 1817.

Many figurative uses are from horseracing or hunting (such as to run (something) into the ground "carry to excess, exhaust by constant pursuit," 1836, American English).

To run across "meet by chance, fall in with" is attested from 1855, American English. To run into in this sense is by 1902. To run around with "consort with" is from 1887.

In reference to fevers by 1918. To run a (red) traffic signal is by 1933. Of tests, experiments, etc., by 1947. Of computers by 1952. Time has been running out since c. 1300. To run in the family is by 1771. The figurative expression run interference (1929) is from U.S. football. To run late is from 1954.

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