in carpentry, "a joint at a 45 degree angle," 1670s, of uncertain origin, perhaps from mitre, via notion of joining of the two peaks of the folded cap. As a verb, to make or join with a miter-joint," from 1731. Related: Mitered. Miter-box is attested from 1670s.
1802, from French fez, from Turkish fes, probably ultimately from Fez, the city in Morocco, where this type of tasseled cap was principally made. Made part of the Turkish official dress by sultan Mahmud II.
a broad-meaning word used of things that move slowly or with difficulty, "of obscure etymology" [OED]. From 1620s as "handle of a pitcher," this sense probably from Scottish lugge "earflap of a cap; ear" (late 15c. and according to OED still the common word for "ear" in 19c. Scotland), which is probably from Scandinavian (compare Swedish lugg "forelock," Norwegian lugg "tuft of hair") and influenced by the verb. The connecting notion is "something that can be gripped and pulled."
Applied 19c. to mechanical objects that can be grabbed or gripped. Meaning "stupid fellow" is from 1924; that of "lout, sponger" is 1931, American English. Compare lug-nut (1869), nut closed at one end as a cap.
c. 1840, type of bonnet formerly worn by Scottish plowmen, from Tam O'Shanter "Tom of Shanter," name of hero in a poem of the same name by Robert Burns, written 1790. The woolen cap became fashionable for ladies c. 1887.
"a vain, shallow fellow, a fop," 1570s, from cokkes comb (1560s, see cockscomb), the name of a device worn in the cap by licensed fools. Johnson has coxcomical (adj.) "foppish, conceited," but discourages it as "a low word unworthy of use."
1650s, from hand in cap, a game whereby two bettors would engage a neutral umpire to determine the odds in an unequal contest. The bettors would put their hands holding forfeit money into a hat or cap. The umpire would announce the odds and the bettors would withdraw their hands — hands full meaning that they accepted the odds and the bet was on, hands empty meaning they did not accept the bet and were willing to forfeit the money. If one forfeited, then the money went to the other. If both agreed either on forfeiting or going ahead with the wager, then the umpire kept the money as payment. The custom, though not the name, is attested from 14c. ("Piers Plowman").
Reference to horse racing is 1754 (Handy-Cap Match), where the umpire decrees the superior horse should carry extra weight as a "handicap;" this led to sense of "encumbrance, disability" first recorded 1890. The main modern sense, "a mental or physical disability," is the last to develop, early 20c.