Etymology
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cat (n.)

Old English catt (c. 700) "domestic cat," from West Germanic (c. 400-450), from Proto-Germanic *kattuz (source also of Old Frisian katte, Old Norse köttr, Dutch kat, Old High German kazza, German Katze), from Late Latin cattus.

The near-universal European word now, it appeared in Europe as Latin catta (Martial, c. 75 C.E.), Byzantine Greek katta (c. 350) and was in general use on the continent by c. 700, replacing Latin feles. Probably ultimately Afro-Asiatic (compare Nubian kadis, Berber kadiska, both meaning "cat"). Arabic qitt "tomcat" may be from the same source. Cats were domestic in Egypt from c. 2000 B.C.E., but not a familiar household animal to classical Greeks and Romans. The nine lives have been proverbial at least since 1560s.

The Late Latin word also is the source of Old Irish and Gaelic cat, Welsh kath, Breton kaz, Italian gatto, Spanish gato, French chat (12c.). Independent, but ultimately from the same source are words in the Slavic group: Old Church Slavonic kotuka, kotel'a, Bulgarian kotka, Russian koška, Polish kot, along with Lithuanian katė and non-Indo-European Finnish katti, which is from Lithuanian.

Extended to lions, tigers, etc. c. 1600. As a term of contempt for a woman, from early 13c. Slang sense of "prostitute" is from at least c. 1400. Slang sense of "fellow, guy," is from 1920, originally in African-American vernacular; narrower sense of "jazz enthusiast" is recorded from 1931.

Cat's paw (1769, but cat's foot in the same sense, 1590s) refers to the old folk tale in which the monkey tricks the cat into pawing chestnuts from a fire; the monkey gets the roasted nuts, the cat gets a burnt paw. Cat burglar is from 1907, so called for stealth. Cat-witted "small-minded, obstinate, and spiteful" (1670s) deserved to survive. For Cat's meow, cat's pajamas, see bee's knees. For let the cat out of the bag, see bag (n.).

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bag (n.)

"small sack," c. 1200, bagge, probably from Old Norse baggi "pack, bundle," or a similar Scandinavian source. OED rejects connection to other Germanic words for "bellows, belly" as without evidence and finds a Celtic origin untenable. In some senses perhaps from Old French bague, which is also from Germanic.

As disparaging slang for "woman" it dates from 1924 in modern use (but various specialized senses of this are much older, and compare baggage). Meaning "person's area of interest or expertise" is 1964, from African-American vernacular, from jazz sense of "category," probably via notion of putting something in a bag. Meaning "fold of loose skin under the eye" is by 1867. Related: bags.

To be left holding the bag (and presumably nothing else), "cheated, swindled" is attested by 1793. Many figurative senses, such as the verb meaning "to kill game" (1814) and its colloquial extension to "catch, seize, steal" (1818) are from the notion of the game bag (late 15c.) into which the product of the hunt was placed. This also probably explains modern slang in the bag "assured, certain" (1922, American English).

To let the cat out of the bag "reveal the secret" is from 1760. The source is probably the French expression Acheter chat en poche "buy a cat in a bag," which is attested in 18c. French and explained in Bailey's "Universal Etymological English Dictionary" (1736), under the entry for To buy a pig in a poke as "to buy a Thing without looking at it, or enquiring into the Value of it." (Similar expressions are found in Italian and German; and in English, Wyclif (late 14c.) has To bye a catte in þo sakke is bot litel charge). Thus to let the cat out of the bag would be to inadvertently reveal the hidden truth of a matter one is attempting to pass off as something better or different, which is in line with the earliest uses in English.

Sir Joseph letteth the cat out of the bag, and sheweth principles inimical to the cause of true philosophy, by wishing to make great men Fellows, instead of wise men ["Peter Pindar," "Peter's Prophecy," 1788]
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bad (adj.)

c. 1300, "inadequate, unsatisfactory, worthless; unfortunate;" late 14c., "wicked, evil, vicious; counterfeit;" from 13c. in surnames (William Badde, Petri Badde, Asketinus Baddecheese, Rads Badinteheved). Rare before 1400, and evil was more common until c. 1700 as the ordinary antithesis of good. It has no apparent relatives in other languages.* Possibly from Old English derogatory term bæddel and its diminutive bædling "effeminate man, hermaphrodite, pederast," which probably are related to bædan "to defile."

The orig. word, AS. bæddel, ME. baddel, on account of its sinister import, is scarcely found in literature, but, like other words of similar sense, it prob. flourished in vulgar speech as an indefinite term of abuse, and at length, divested of its original meaning, emerged in literary use as a mere adj., badde, equiv. to the older evil. [Century Dictionary, 1897]

Comparable words in the other Indo-European languages tend to have grown from descriptions of specific qualities, such as "ugly," "defective," "weak," "faithless," "impudent," "crooked," "filthy" (such as Greek kakos, probably from the word for "excrement;" Russian plochoj, related to Old Church Slavonic plachu "wavering, timid;" Persian gast, Old Persian gasta-, related to gand "stench;" German schlecht, originally "level, straight, smooth," whence "simple, ordinary," then "bad").

Comparative and superlative forms badder, baddest were common 14c.-18c. and used as recently as Defoe (but not by Shakespeare), but yielded to comparative worse and superlative worst (which had belonged to evil and ill).

Meaning "uncomfortable, sorry" is 1839, American English colloquial. To go bad "putrefy" is from 1884. Not bad "fairly good" is by 1771. Ironic use as a word of approval is said to be at least since 1890s orally, originally in African-American vernacular, emerging in print 1928 in a jazz context. It might have emerged from the ambivalence of expressions like bad nigger, used as a term of reproach by whites, but among blacks sometimes representing one who stood up to injustice, but in the U.S. West bad man also had a certain ambivalence:

These are the men who do most of the killing in frontier communities, yet it is a noteworthy fact that the men who are killed generally deserve their fate. [Farmer and Henley, "Slang and Its Analogues"]

*Persian has bad in more or less the same sense as the English word, but this is regarded by linguists as a coincidence. The forms of the words diverge as they are traced back in time (Persian bad comes from Middle Persian vat), and such accidental convergences exist across many languages, given the vast number of words in each and the limited range of sounds humans can make to signify them. Among other coincidental matches with English are Korean mani "many," Chinese pei "pay," Nahuatl (Aztecan) huel "well," Maya hol "hole."

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