Etymology
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Words related to *skel-

coulter (n.)

also colter, "iron blade or sharp-edged wheel attached to the beam of a plow to cut the ground," Old English culter, from Latin culter "a knife, iron blade in a plowshare," from PIE root *skel- (1) "to cut." As a surname (13c.), probably from Coulter in Lancashire.

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cutlass (n.)

"short sword or large knife with a flat, wide, slightly curved blade," used for cutting more than thrusting, 1590s, from French coutelas (16c.), which is probably from Italian coltellaccio "large knife," with augmentative suffix -accio + coltello "knife," from Latin cultellus "small knife," diminutive of culter "knife, plowshare," from PIE *kel-tro-, suffixed form of root *skel- (1) "to cut." Not related to cut.

half 

Old English half, halb (Mercian), healf (W. Saxon) "side, part," not necessarily of equal division (original sense preserved in behalf), from Proto-Germanic *halba- "something divided" (source also of Old Saxon halba, Old Norse halfr, Old Frisian, Middle Dutch half, German halb, Gothic halbs "half"), a word of no certain etymology. Perhaps from PIE root *skel- (1) "to cut," or perhaps a substratum word. Noun, adjective, and adverb all were in Old English.

Used also in Old English phrases, as in modern German, to mean "one half unit less than," for example þridda healf "two and a half," literally "half third." The construction in two and a half, etc., is first recorded c. 1200. Of time, in half past ten, etc., first attested 1750; in Scottish, the half often is prefixed to the following hour (as in German, halb elf = "ten thirty").

To go off half-cocked in the figurative sense "speak or act too hastily" (1833) is in allusion to firearms going off prematurely; half-cocked in a literal sense "with the cock lifted to the first catch, at which position the trigger does not act" is recorded by 1750. In 1770 it was noted as a synonym for "drunk." Bartlett ("Dictionary of Americanisms," 1848) writes that it was "a metaphorical expression borrowed from the language of sportsmen, and is applied to a person who attempts a thing in a hurry without due preparation, and consequently fails."

halve (v.)
Middle English halven, halfen "to divide in halves" (c. 1200), from half (n.). Meaning "to reduce by half" is from c. 1400. Related: Halved; halving.
scale (n.1)

[one of the skin plates on fish or snakes] c. 1300, from Old French escale "cup, scale, shell pod, husk" (12c., Modern French écale), from Frankish *skala or some other Germanic source, from Proto-Germanic *skæla "to split, divide" (source also of Dutch schaal "a scale, husk," Old High German scala "shell," Gothic skalja "tile," Old English scealu "shell, husk"), from PIE root *skel- (1) "to cut." A prehistoric cognate of scale (n.2) "weighing instrument."

In reference to humans, as a condition of certain skin diseases, it is attested from late 14c. Extended in botany to coverings of leaf-buds, etc., by 1776. As what falls from one's eye when blindness ends (usually figurative), it echoes Acts ix:18 (Latin tanquam squamæ, Greek hosei lepides).

scale (n.2)

[weighing instrument] early 15c., extended to the whole instrument from the earlier sense of "pan of a balance" (late 14c.); earlier still "drinking cup" (c. 1200), from Old Norse skal "bowl, drinking cup," in plural, "weighing scale."

This is from a noun derivative of Proto-Germanic *skæla "to split, divide" (source also of Old Norse skel "shell," Old English scealu, Old Saxon skala "a bowl (to drink from)," Old High German scala, German Schale "a bowl, dish, cup," Middle Dutch scale, Dutch schaal "drinking cup, bowl, shell, scale of a balance"), from PIE root *skel- (1) "to cut."

The connecting sense seems to be of half of a bivalve ("split") shell used as a drinking cup or a pan for weighing; compare scallop, which is from the same root. But according to Paulus Diaconus the "drinking cup" sense originated from a supposed custom of making goblets from skulls (see skull). Scales as a name for the zodiac constellation Libra is attested in English from 1630s.

scalene (adj.)

"having unequal sides," in mathematics, 1680s, from Late Latin scalenus, from Greek skalenōs "uneven, unequal, odd (numbered)," as a noun, "triangle with unequal sides" (trigōnon skalēnon), from skallein "to chop, hoe, dig, stir up" (compare its derivative, skalops "a mole"), from PIE root *skel- (1) "to cut."

scallop (n.)

type of edible bivalve mollusk, mid-14c., scalop, from Old French escalope "shell (of a nut), carpace," a variant of eschalope, which probably is from a Germanic source (compare Old Norse skalpr "sheath," Middle Dutch schelpe "shell"), from PIE root *skel- (1) "to cut."

Extended 17c. to objects shaped or ornaments cut like scallop shells, especially in design and dress. The shells of the larger species, often colorfully marked, have been used as domestic utensils. It also was a symbol of St. James the Great, and the shells were worn or carried as by pilgrims who had been to his shrine in Compostella.

scalp (n.)

mid-14c. (c. 1200 as a surname), "crown or top of the head (including hair)," presumably from a Scandinavian source (though exact sense cognates are wanting) related to Old Norse skalli "a bald head," skalpr "sheath, scabbard," from PIE root *skel- (1) "to cut," which is also the base of shell, skull, and scallop.

French scalpe, German, Danish, Swedish skalp are from English. Meaning "head skin and hair cut or torn from the head as a victory trophy" is from c. 1600, in Holland's Pliny, 1670s in reference to some North American tribal customs; as proof of the killing of an animal by 1703. Figuratively, as a symbol of victory, by 1757.

scalpel (n.)

"small, light surgical knife held like a pen," 1742, from Latin scalpellum "a surgical knife," diminutive of scalprum "knife, chisel, tool for scraping or cutting," from scalpere "to carve, cut," which is related to sculpere "to carve" (from PIE root *skel- (1) "to cut").