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Old English wis, from Proto-Germanic *wisaz (cf. Old Saxon, Old Frisian wis, Old Norse viss, Dutch wijs, German weise "wise"), from past participle adjective *wittos of PIE root *weid- "to see," hence "to know" (see vision). Slang meaning "aware, cunning" first attested 1896. Related to the source of Old English witan "to know, wit."
A wise man has no extensive knowledge; He who has extensive knowledge is not a wise man. [Lao-tzu, "Tao te Ching," c.550 B.C.E.]Wise guy is attested from 1896, American English. Wisenheimer, with mock German or Yiddish surname suffix, first recorded 1904.
"way of proceeding, manner," Old English wise, ultimately from the same source as wise (adj.). Cf. Old Saxon wisa, Old Frisian wis, Danish vis, Middle Dutch wise, Dutch wijs, Old High German wisa, German Weise "way, manner." Most common in English now as a suffix (e.g. likewise). For sense evolution from "to see" to "way of proceeding," cf. cognate Greek eidos "form, shape, kind," also "course of action." Ground sense is "to see/know the way."
To become aware of; discover: Had you gotten wise to me, or was it an accident? (1896+)