dumb
O.E. dumb "silent, unable to speak," from PIE *dheubh- "confusion, stupefaction, dizziness." O.E., Goth. (thumb) and O.N. (dumbr) forms meant only "mute, speechless;" in O.H.G. (thumb) it meant both this and "stupid," and in Mod.Ger. this latter became the only sense. Meaning "foolish, ignorant" was occasionally in Eng. from c.1323, but modern use (1823) comes from infl. of Ger. dumm. Applied to silent contrivances, hence dumbwaiter (1749). To dumb down is from 1933.
mute
c.1374, mewet, "silent," from O.Fr. muet, dim. of mut, mo, from L. mutus "silent, dumb," probably from imitative base *mu- (cf. Skt. mukah "dumb," Gk. myein "to be shut," of the mouth). Assimilated in form in 16c. to L. mutus. The verb is first attested 1861. Musical noun sense first recorded 1811, of stringed instruments, 1841, of horns.