archyoudaoicibaDictYouDict[arch 词源字典]
arch: [14] English acquired arch via Old French arche and a hypothetical Vulgar Latin *arca from Latin arcus ‘curve, arch, bow’ (from which English also got arc [14]). When it first came into the language it was still used in the general sense of ‘curve, arc’ as well as ‘curved structure’ (Chaucer in his Treatise on the astrolabe 1391 wrote of ‘the arch of the day … from the sun arising till it go to rest’), but this had died out by the mid 19th century.

Vulgar Latin *arca also produced Italian arcata, which entered English via French as arcade in the 18th century. Arch meaning ‘saucy’ is an adjectival use of the prefix arch- (as in archetype).
=> arc[arch etymology, arch origin, 英语词源]
arch (n.)youdaoicibaDictYouDict
c. 1300, from Old French arche "arch of a bridge" (12c.), from Latin arcus "a bow" (see arc). Replaced native bow (n.1). Originally architectural in English; transferred by early 15c. to anything having this form (eyebrows, etc.).
arch (adj.)youdaoicibaDictYouDict
1540s, "chief, principal," from prefix arch-; used in 12c. archangel, etc., but extended to so many derogatory uses (arch-rogue, arch-knave, etc.) that by mid-17c. it acquired a meaning of "roguish, mischievous," since softened to "saucy." Also found in archwife (late 14c.), variously defined as "a wife of a superior order" or "a dominating woman, virago."
arch (v.)youdaoicibaDictYouDict
early 14c., "to form an arch" (implied in arched); c. 1400, "to furnish with an arch," from arch (n.). Related: Arching.