1520s, "to overcome in argument," from Latin convincere "to overcome decisively," from assimilated form of com-, here probably an intensive prefix (see com-), + vincere "to conquer" (from nasalized form of PIE root *weik- (3) "to fight, conquer"). Meaning "to firmly persuade or satisfy by argument or evidence" is from c. 1600. Related: Convinced; convincing; convincingly.
To convince a person is to satisfy his understanding as to the truth of a certain statement; to persuade him is, by derivation, to affect his will by motives; but it has long been used also for convince, as in Luke xx. 6, "they be persuaded that John was a prophet." There is a marked tendency now to confine persuade to its own distinctive meaning. [Century Dictionary, 1897]
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