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Old English ræcan, reccan "reach out, stretch out, extend, hold forth," also "succeed in touching, succeed in striking; address, speak to," also "offer, present, give, grant," from West Germanic *raikjan "stretch out the hand" (cf. Old Frisian reka, Middle Dutch reiken, Dutch reiken, Old High German and German reichen), from Proto-Germanic *raikijanau, perhaps from PIE root *reig- "to stretch out" (cf. Sanskrit rjyati "he stretches himself," riag "torture" (by racking); Greek oregein "to reach, extend;" Lithuanian raižius "to stretch oneself;" Old Irish rigim "I stretch").
Shakespeare uses the now-obsolete past tense form raught (Old English ræhte). Meaning "arrive at" is early 14c.; that of "succeed in influencing" is from 1660s. Related: Reached; reaching. Reach-me-down "ready-made" (of clothes) is recorded from 1862, from notion of being on the rack in a finished state.
1520s, from reach (v.); earliest use is of stretches of water. Meaning "extent of reaching" is from 1540s; that of "act of reaching" is from 1560s.
Ah, but a man's reach should exceed his grasp,
Or what's a heaven for?
[Browning, "Andrea del Sarto"]