Etymology
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roach (n.1)

a shortened form of cockroach, on the mistaken notion that it is a compound, attested by 1830.

In contemporary writing the shortening sometimes is credited to a polite desire to avoid the sexual connotation in the first syllable of the full word, especially among Americans, but this seems to be another English fiction and early uses typically are in natural history publications.

The Translator must ask pardon of any American lady, into whose hands this book may by chance fall, for making use of so vulgar a term. "Cock-roaches" in the United States, as we are told by one of the numerous English travellers through that country, are always called "roaches" by the fair sex, for the sake of euphony. [B.D. Walsh, footnote in translation of "The Acharnians," 1848] 

The meaning "butt of a marijuana cigarette" is recorded by 1938, perhaps from resemblance to the insect but rather this might be a different word entirely. Related: Roach-clip (by 1968).

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roach (n.2)

common small freshwater fish of northern Europe, late 12c., from Old French roche (13c.), a name of uncertain origin, perhaps from a Germanic source (compare Middle Dutch roch, Low German ruche). Applied later to fish in North America that resemble it.

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rooster (n.)

"cock, male of the domestic hen," 1772, agent noun from roost (v.); earlier roost cock, c. 1600, in sense of "the roosting bird." Said to have become favored in the U.S. (it was noted by 1836 as a Yankeeism) and said to have been originally a puritan alternative to cock (n.) after that word had acquired the secondary sense "penis" (and compare roach).

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cockroach (n.)

popular name of a troublesome, voracious insect genus, 1620s, folk etymology (as if from cock (n.1) + roach; compare cockchafer) of Spanish cucaracha "chafer, beetle," from cuca "kind of caterpillar." Folk etymology also holds that the first element is from caca "excrement," perhaps because of the insect's offensive smell.

A certaine India Bug, called by the Spaniards a Cacarootch, the which creeping into Chests they eat and defile with their ill-sented dung [Capt. John Smith, "Virginia," 1624].
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cut (v.)

c. 1300, "to make, with an edged tool or instrument, an incision in; make incisions for the purpose of dividing into two or more parts; remove by means of a cutting instrument;" of an implement, "have a cutting edge," according to Middle English Compendium from a presumed Old English *cyttan, "since ME has the normal regional variants of the vowel." Others suggest a possible Scandinavian etymology from North Germanic *kut- (source also of Swedish dialectal kuta "to cut," kuta "knife," Old Norse kuti "knife"), or that it is from Old French couteau "knife."

It has largely displaced Old English ceorfan (see carve (v.)), snian, and scieran (see shear). The past participle is also cut, though cutted sometimes has been used since Middle English.

From early 14c. as "to make or fashion by cutting or carving." From c. 1400 as "to intersect or cross." From early 15c. as "abridge or shorten by omitting a part."

Meaning "to wound the sensibilities of" is from 1580s (to cut the heart in the same sense is attested from early 14c.). Sense of "sever connection or relations with" is from 1630s.

Meaning "to be absent without excuse" is British university slang from 1794. Colloquial or slang sense of "move off with directness and rapidity" is from 1580s. Meaning "divide (a deck of cards) at random into parts before the deal" to prevent cheating is from 1530s.

Meaning "to dilute, adulterate" (liquor, etc.) is by 1930. Colloquial sense of "to divide or share" is by 1928, perhaps an image from meat-carving at table. As a director's call to halt recording or performing, by 1931 (in an article about Pete, the bulldog with the black-ringed eye in the Hal Roach studios shorts, who was said to know the word). The sense of "perform, execute" (c. 1600) is in cut capers "frisk about;" cut a dash "make a display."

To cut down is from late 14c. as "to fell;" by 1821 as "to slay" (as with a sword); 1857 as "to curtail." To cut (someone or something) down to size is from 1821 as "reduce to suitable dimensions;" the figurative sense, "reduce to the proper level of importance," is by 1927.

To cut in "enter suddenly and unceremoniously" is from 1610s; sense of "suddenly join in conversation, interrupt" is by 1830. To cut up "cut in pieces" is from 1570s. To cut back is from 1871 as "prune by cutting off shoots," 1913 in cinematography, "return to a previous scene by repeating a part of it," 1943 as "reduce, decrease" (of expenditures, etc.). To cut (something) short "abridge, curtail, interrupt" is from 1540s.

In nautical use to cut a feather (1620s) is to move so fast as to make water foam under the bow. To cut and run (1704) also is originally nautical, "cut cable and set sail immediately," as in an emergency, hence, generally, "to make off suddenly."

To cut the teeth "have the teeth grow through the gums" as an infant is from 1670s. To cut both ways in the figurative sense of "have a good and bad effect" is from c. 1600. To cut loose "set (something) free" is by 1828; intransitive sense "begin to act freely" is by 1909.

Cut it out "remove (something) by or as if by cutting" yielded a figurative use in the command cut it out! "Stop! That's enough!" by 1933. The evolution seems to have begun earlier. A piece attributed to the Chicago Live Stock World that made the rounds in trade publications 1901-02 begins:

When you get 'hot' about something and vow you are going to rip something or somebody up the back—cut it out.
If you feel disposed to try the plan of building yourself up by tearing some one else down—cut it out.

Playing on both senses, it ends with "Should you, after reading this preachy stuff, fear you might forget some of the good advice—cut it out."

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